A limit describes the behavior of a function as the input values get close to a specific value. We write [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)=L[\/latex] to indicate that as [latex]x[\/latex] approaches [latex]a[\/latex], the output value [latex]f(x)[\/latex] approaches [latex]L[\/latex].<\/p>\r\n
The limit of a function as [latex]x[\/latex] approaches [latex]a[\/latex] can exist even when [latex]f(a)[\/latex] does not exist or when [latex]f(a)\\neq L[\/latex]. The value of the limit is not affected by the output value of [latex]f(x)[\/latex] at [latex]a[\/latex].<\/p>\r\n
One-sided limits:<\/strong><\/p>\r\n\r\n Two-sided limits:<\/strong> A two-sided limit exists if and only if both one-sided limits exist and are equal: [latex]\\underset{x\\to a^{-}}{\\mathrm{lim}}f(x)=\\underset{x\\to a^{+}}{\\mathrm{lim}}f(x)[\/latex]. When we refer to \"a limit\" without specifying, we mean a two-sided limit.<\/p>\r\n Finding limits using graphs:<\/strong> A graph provides a visual method for determining limits. If the function has a limit as [latex]x[\/latex] approaches [latex]a[\/latex], the branches of the graph will approach the same y-coordinate near [latex]x=a[\/latex] from both the left and right.<\/p>\r\n\r\n Finding limits using tables:<\/strong> A table can be used to determine if a function has a limit by showing input values that approach [latex]a[\/latex] from both directions.<\/p>\r\n\r\n When limits exist and are finite, we can perform operations using the properties below. Let [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)=A[\/latex] and [latex]\\underset{x\\to a}{\\mathrm{lim}}g(x)=B[\/latex].<\/p>\r\n Basic properties:<\/strong><\/p>\r\n\r\n Limits of polynomials:<\/strong> For a polynomial function [latex]p(x)[\/latex], we have [latex]\\underset{x\\to a}{\\mathrm{lim}}p(x)=p(a)[\/latex]. The limit can be found by finding the sum of the limits of the individual terms, or more simply, by direct substitution of [latex]a[\/latex] into the polynomial.<\/p>\r\n Evaluating limits of quotients:<\/strong> When the denominator evaluates to 0, rewrite the function algebraically before applying limit properties:<\/p>\r\n\r\n A function is continuous at [latex]x=a[\/latex] if there are no holes or breaks in its graph at that point. A discontinuous function is one that has any hole or break in its graph.<\/p>\r\n Three conditions for continuity:<\/strong> A function [latex]f(x)[\/latex] is continuous at [latex]x=a[\/latex] if all three conditions hold:<\/p>\r\n\r\n If any condition fails, the function is discontinuous at [latex]x=a[\/latex].<\/p>\r\n Functions that are continuous everywhere:<\/strong><\/p>\r\n\r\n Functions continuous on their domain:<\/strong><\/p>\r\n\r\n Types of discontinuities:<\/strong><\/p>\r\n\r\n Determining continuity of piecewise functions:<\/strong><\/p>\r\n To check continuity at a boundary point [latex]x=a[\/latex]:<\/p>\r\n\r\n The derivative of a function at a point represents the instantaneous rate of change at that point. It can also be interpreted as the slope of the tangent line to the graph at that point.<\/p>\r\n Instantaneous rate of change<\/strong>: The derivative measures how quickly the output of a function is changing at a specific input value, unlike average rate of change which measures change over an interval.<\/p>\r\n Tangent line<\/strong>: A line that touches the graph of a function at exactly one point and has the same slope as the function at that point. The equation of the tangent line to [latex]f(x)[\/latex] at [latex]x=a[\/latex] can be found using the derivative and point-slope form.<\/p>\r\n\r\n continuous function<\/strong><\/p>\r\n A function that has no holes or breaks in its graph; a function [latex]f(x)[\/latex] is continuous at [latex]x=a[\/latex] if [latex]f(a)[\/latex] exists, [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)[\/latex] exists, and [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)=f(a)[\/latex].<\/p>\r\n derivative<\/strong><\/p>\r\n The instantaneous rate of change of a function at a point; also represents the slope of the tangent line to the graph of the function at that point.<\/p>\r\n discontinuous function<\/strong><\/p>\r\n A function that is not continuous at [latex]x=a[\/latex]; a function that has one or more holes or breaks in its graph.<\/p>\r\n infinite discontinuity<\/strong><\/p>\r\n A point of discontinuity in a function [latex]f(x)[\/latex] at [latex]x=a[\/latex] where one or both of the one-sided limits approaches positive or negative infinity; the graph typically has a vertical asymptote at [latex]x=a[\/latex].<\/p>\r\n instantaneous rate of change<\/strong><\/p>\r\n The rate at which a function is changing at a specific point, measured by the derivative; describes how quickly the output changes relative to the input at an exact moment.<\/p>\r\n jump discontinuity<\/strong><\/p>\r\n A point of discontinuity in a function [latex]f(x)[\/latex] at [latex]x=a[\/latex] where both the left-hand and right-hand limits exist, but [latex]\\underset{x\\to a^{-}}{\\mathrm{lim}}f(x)\\neq\\underset{x\\to a^{+}}{\\mathrm{lim}}f(x)[\/latex]; the graph \"jumps\" from one value to another.<\/p>\r\n left-hand limit<\/strong><\/p>\r\n The limit of values of [latex]f(x)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] from the left, denoted [latex]\\underset{x\\to a^{-}}{\\mathrm{lim}}f(x)=L[\/latex]. The values of [latex]f(x)[\/latex] can get as close to the limit [latex]L[\/latex] as we like by taking values of [latex]x[\/latex] sufficiently close to [latex]a[\/latex] such that [latex]x<a[\/latex] and [latex]x\\neq a[\/latex].<\/p>\r\n limit<\/strong><\/p>\r\n When it exists, the value [latex]L[\/latex] that the output of a function [latex]f(x)[\/latex] approaches as the input [latex]x[\/latex] gets closer and closer to [latex]a[\/latex] but does not equal [latex]a[\/latex]. The value of the output [latex]f(x)[\/latex] can get as close to [latex]L[\/latex] as we choose by using input values of [latex]x[\/latex] sufficiently near to [latex]x=a[\/latex], but not necessarily at [latex]x=a[\/latex]. Denoted [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)=L[\/latex].<\/p>\r\n properties of limits<\/strong><\/p>\r\n A collection of theorems for finding limits of functions by performing mathematical operations on the limits.<\/p>\r\n removable discontinuity<\/strong><\/p>\r\n A point of discontinuity in a function [latex]f(x)[\/latex] at [latex]x=a[\/latex] where the limit exists but either [latex]f(a)[\/latex] does not exist or [latex]f(a)\\neq\\underset{x\\to a}{\\mathrm{lim}}f(x)[\/latex]; the function can be redefined at its discontinuous point to make it continuous.<\/p>\r\n right-hand limit<\/strong><\/p>\r\n The limit of values of [latex]f(x)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] from the right, denoted [latex]\\underset{x\\to a^{+}}{\\mathrm{lim}}f(x)=L[\/latex]. The values of [latex]f(x)[\/latex] can get as close to the limit [latex]L[\/latex] as we like by taking values of [latex]x[\/latex] sufficiently close to [latex]a[\/latex] where [latex]x>a[\/latex] and [latex]x\\neq a[\/latex].<\/p>\r\n stepwise function<\/strong><\/p>\r\n A function that remains constant over intervals and then jumps instantaneously to different values; an example of a function with jump discontinuities.<\/p>\r\n tangent line<\/strong><\/p>\r\n A line that touches the graph of a function at exactly one point and has the same instantaneous rate of change (slope) as the function at that point.<\/p>\r\n two-sided limit<\/strong><\/p>\r\n The limit of a function [latex]f(x)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] is equal to [latex]L[\/latex], denoted [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)=L[\/latex], if and only if [latex]\\underset{x\\to a^{-}}{\\mathrm{lim}}f(x)=\\underset{x\\to a^{+}}{\\mathrm{lim}}f(x)[\/latex].<\/p>","rendered":" A limit describes the behavior of a function as the input values get close to a specific value. We write [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)=L[\/latex] to indicate that as [latex]x[\/latex] approaches [latex]a[\/latex], the output value [latex]f(x)[\/latex] approaches [latex]L[\/latex].<\/p>\n The limit of a function as [latex]x[\/latex] approaches [latex]a[\/latex] can exist even when [latex]f(a)[\/latex] does not exist or when [latex]f(a)\\neq L[\/latex]. The value of the limit is not affected by the output value of [latex]f(x)[\/latex] at [latex]a[\/latex].<\/p>\n One-sided limits:<\/strong><\/p>\n Two-sided limits:<\/strong> A two-sided limit exists if and only if both one-sided limits exist and are equal: [latex]\\underset{x\\to a^{-}}{\\mathrm{lim}}f(x)=\\underset{x\\to a^{+}}{\\mathrm{lim}}f(x)[\/latex]. When we refer to “a limit” without specifying, we mean a two-sided limit.<\/p>\n Finding limits using graphs:<\/strong> A graph provides a visual method for determining limits. If the function has a limit as [latex]x[\/latex] approaches [latex]a[\/latex], the branches of the graph will approach the same y-coordinate near [latex]x=a[\/latex] from both the left and right.<\/p>\n Finding limits using tables:<\/strong> A table can be used to determine if a function has a limit by showing input values that approach [latex]a[\/latex] from both directions.<\/p>\n When limits exist and are finite, we can perform operations using the properties below. Let [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)=A[\/latex] and [latex]\\underset{x\\to a}{\\mathrm{lim}}g(x)=B[\/latex].<\/p>\n Basic properties:<\/strong><\/p>\n Limits of polynomials:<\/strong> For a polynomial function [latex]p(x)[\/latex], we have [latex]\\underset{x\\to a}{\\mathrm{lim}}p(x)=p(a)[\/latex]. The limit can be found by finding the sum of the limits of the individual terms, or more simply, by direct substitution of [latex]a[\/latex] into the polynomial.<\/p>\n Evaluating limits of quotients:<\/strong> When the denominator evaluates to 0, rewrite the function algebraically before applying limit properties:<\/p>\n A function is continuous at [latex]x=a[\/latex] if there are no holes or breaks in its graph at that point. A discontinuous function is one that has any hole or break in its graph.<\/p>\n Three conditions for continuity:<\/strong> A function [latex]f(x)[\/latex] is continuous at [latex]x=a[\/latex] if all three conditions hold:<\/p>\n If any condition fails, the function is discontinuous at [latex]x=a[\/latex].<\/p>\n Functions that are continuous everywhere:<\/strong><\/p>\n Functions continuous on their domain:<\/strong><\/p>\n Types of discontinuities:<\/strong><\/p>\n Determining continuity of piecewise functions:<\/strong><\/p>\n To check continuity at a boundary point [latex]x=a[\/latex]:<\/p>\n The derivative of a function at a point represents the instantaneous rate of change at that point. It can also be interpreted as the slope of the tangent line to the graph at that point.<\/p>\n Instantaneous rate of change<\/strong>: The derivative measures how quickly the output of a function is changing at a specific input value, unlike average rate of change which measures change over an interval.<\/p>\n Tangent line<\/strong>: A line that touches the graph of a function at exactly one point and has the same slope as the function at that point. The equation of the tangent line to [latex]f(x)[\/latex] at [latex]x=a[\/latex] can be found using the derivative and point-slope form.<\/p>\n continuous function<\/strong><\/p>\n A function that has no holes or breaks in its graph; a function [latex]f(x)[\/latex] is continuous at [latex]x=a[\/latex] if [latex]f(a)[\/latex] exists, [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)[\/latex] exists, and [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)=f(a)[\/latex].<\/p>\n derivative<\/strong><\/p>\n The instantaneous rate of change of a function at a point; also represents the slope of the tangent line to the graph of the function at that point.<\/p>\n discontinuous function<\/strong><\/p>\n A function that is not continuous at [latex]x=a[\/latex]; a function that has one or more holes or breaks in its graph.<\/p>\n infinite discontinuity<\/strong><\/p>\n A point of discontinuity in a function [latex]f(x)[\/latex] at [latex]x=a[\/latex] where one or both of the one-sided limits approaches positive or negative infinity; the graph typically has a vertical asymptote at [latex]x=a[\/latex].<\/p>\n instantaneous rate of change<\/strong><\/p>\n The rate at which a function is changing at a specific point, measured by the derivative; describes how quickly the output changes relative to the input at an exact moment.<\/p>\n jump discontinuity<\/strong><\/p>\n A point of discontinuity in a function [latex]f(x)[\/latex] at [latex]x=a[\/latex] where both the left-hand and right-hand limits exist, but [latex]\\underset{x\\to a^{-}}{\\mathrm{lim}}f(x)\\neq\\underset{x\\to a^{+}}{\\mathrm{lim}}f(x)[\/latex]; the graph “jumps” from one value to another.<\/p>\n left-hand limit<\/strong><\/p>\n The limit of values of [latex]f(x)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] from the left, denoted [latex]\\underset{x\\to a^{-}}{\\mathrm{lim}}f(x)=L[\/latex]. The values of [latex]f(x)[\/latex] can get as close to the limit [latex]L[\/latex] as we like by taking values of [latex]x[\/latex] sufficiently close to [latex]a[\/latex] such that [latex]x limit<\/strong><\/p>\n When it exists, the value [latex]L[\/latex] that the output of a function [latex]f(x)[\/latex] approaches as the input [latex]x[\/latex] gets closer and closer to [latex]a[\/latex] but does not equal [latex]a[\/latex]. The value of the output [latex]f(x)[\/latex] can get as close to [latex]L[\/latex] as we choose by using input values of [latex]x[\/latex] sufficiently near to [latex]x=a[\/latex], but not necessarily at [latex]x=a[\/latex]. Denoted [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)=L[\/latex].<\/p>\n properties of limits<\/strong><\/p>\n A collection of theorems for finding limits of functions by performing mathematical operations on the limits.<\/p>\n removable discontinuity<\/strong><\/p>\n A point of discontinuity in a function [latex]f(x)[\/latex] at [latex]x=a[\/latex] where the limit exists but either [latex]f(a)[\/latex] does not exist or [latex]f(a)\\neq\\underset{x\\to a}{\\mathrm{lim}}f(x)[\/latex]; the function can be redefined at its discontinuous point to make it continuous.<\/p>\n right-hand limit<\/strong><\/p>\n The limit of values of [latex]f(x)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] from the right, denoted [latex]\\underset{x\\to a^{+}}{\\mathrm{lim}}f(x)=L[\/latex]. The values of [latex]f(x)[\/latex] can get as close to the limit [latex]L[\/latex] as we like by taking values of [latex]x[\/latex] sufficiently close to [latex]a[\/latex] where [latex]x>a[\/latex] and [latex]x\\neq a[\/latex].<\/p>\n stepwise function<\/strong><\/p>\n A function that remains constant over intervals and then jumps instantaneously to different values; an example of a function with jump discontinuities.<\/p>\n tangent line<\/strong><\/p>\n A line that touches the graph of a function at exactly one point and has the same instantaneous rate of change (slope) as the function at that point.<\/p>\n two-sided limit<\/strong><\/p>\n The limit of a function [latex]f(x)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] is equal to [latex]L[\/latex], denoted [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)=L[\/latex], if and only if [latex]\\underset{x\\to a^{-}}{\\mathrm{lim}}f(x)=\\underset{x\\to a^{+}}{\\mathrm{lim}}f(x)[\/latex].<\/p>\n","protected":false},"author":67,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":263,"module-header":"cheat_sheet","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/716"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/716\/revisions"}],"predecessor-version":[{"id":5425,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/716\/revisions\/5425"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/263"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/716\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=716"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=716"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=716"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=716"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\r\n \t
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Properties of Limits<\/h3>\r\n
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Continuity<\/h3>\r\n
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Derivatives<\/h3>\r\n
Key Equations<\/h2>\r\n
\r\n\r\n
\r\n Limit notation<\/strong><\/td>\r\n [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)=L[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Left-hand limit<\/strong><\/td>\r\n [latex]\\underset{x\\to a^{-}}{\\mathrm{lim}}f(x)=L[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Right-hand limit<\/strong><\/td>\r\n [latex]\\underset{x\\to a^{+}}{\\mathrm{lim}}f(x)=L[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Limit of a constant<\/strong><\/td>\r\n [latex]\\underset{x\\to a}{\\mathrm{lim}}k=k[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Limit of constant times a function<\/strong><\/td>\r\n [latex]\\underset{x\\to a}{\\mathrm{lim}}[k\\cdot f(x)]=k\\underset{x\\to a}{\\mathrm{lim}}f(x)[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Limit of a sum<\/strong><\/td>\r\n [latex]\\underset{x\\to a}{\\mathrm{lim}}[f(x)+g(x)]=\\underset{x\\to a}{\\mathrm{lim}}f(x)+\\underset{x\\to a}{\\mathrm{lim}}g(x)[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Limit of a difference<\/strong><\/td>\r\n [latex]\\underset{x\\to a}{\\mathrm{lim}}[f(x)-g(x)]=\\underset{x\\to a}{\\mathrm{lim}}f(x)-\\underset{x\\to a}{\\mathrm{lim}}g(x)[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Limit of a product<\/strong><\/td>\r\n [latex]\\underset{x\\to a}{\\mathrm{lim}}[f(x)\\cdot g(x)]=\\underset{x\\to a}{\\mathrm{lim}}f(x)\\cdot\\underset{x\\to a}{\\mathrm{lim}}g(x)[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Limit of a quotient<\/strong><\/td>\r\n [latex]\\underset{x\\to a}{\\mathrm{lim}}\\frac{f(x)}{g(x)}=\\frac{\\underset{x\\to a}{\\mathrm{lim}}f(x)}{\\underset{x\\to a}{\\mathrm{lim}}g(x)}[\/latex], where [latex]\\underset{x\\to a}{\\mathrm{lim}}g(x)\\neq 0[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Limit of a power<\/strong><\/td>\r\n [latex]\\underset{x\\to a}{\\mathrm{lim}}[f(x)]^{n}=[\\underset{x\\to a}{\\mathrm{lim}}f(x)]^{n}[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Limit of a root<\/strong><\/td>\r\n [latex]\\underset{x\\to a}{\\mathrm{lim}}\\sqrt[n]{f(x)}=\\sqrt[n]{\\underset{x\\to a}{\\mathrm{lim}}f(x)}[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Limit of a polynomial<\/strong><\/td>\r\n [latex]\\underset{x\\to a}{\\mathrm{lim}}p(x)=p(a)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n Glossary<\/h2>\r\n
Essential Concepts<\/h2>\n
Finding Limits<\/h3>\n
\n
\n
\n
Properties of Limits<\/h3>\n
\n
\n
Continuity<\/h3>\n
\n
\n
\n
\n
\n
Derivatives<\/h3>\n
Key Equations<\/h2>\n
\n\n
\n Limit notation<\/strong><\/td>\n [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)=L[\/latex]<\/td>\n<\/tr>\n \n Left-hand limit<\/strong><\/td>\n [latex]\\underset{x\\to a^{-}}{\\mathrm{lim}}f(x)=L[\/latex]<\/td>\n<\/tr>\n \n Right-hand limit<\/strong><\/td>\n [latex]\\underset{x\\to a^{+}}{\\mathrm{lim}}f(x)=L[\/latex]<\/td>\n<\/tr>\n \n Limit of a constant<\/strong><\/td>\n [latex]\\underset{x\\to a}{\\mathrm{lim}}k=k[\/latex]<\/td>\n<\/tr>\n \n Limit of constant times a function<\/strong><\/td>\n [latex]\\underset{x\\to a}{\\mathrm{lim}}[k\\cdot f(x)]=k\\underset{x\\to a}{\\mathrm{lim}}f(x)[\/latex]<\/td>\n<\/tr>\n \n Limit of a sum<\/strong><\/td>\n [latex]\\underset{x\\to a}{\\mathrm{lim}}[f(x)+g(x)]=\\underset{x\\to a}{\\mathrm{lim}}f(x)+\\underset{x\\to a}{\\mathrm{lim}}g(x)[\/latex]<\/td>\n<\/tr>\n \n Limit of a difference<\/strong><\/td>\n [latex]\\underset{x\\to a}{\\mathrm{lim}}[f(x)-g(x)]=\\underset{x\\to a}{\\mathrm{lim}}f(x)-\\underset{x\\to a}{\\mathrm{lim}}g(x)[\/latex]<\/td>\n<\/tr>\n \n Limit of a product<\/strong><\/td>\n [latex]\\underset{x\\to a}{\\mathrm{lim}}[f(x)\\cdot g(x)]=\\underset{x\\to a}{\\mathrm{lim}}f(x)\\cdot\\underset{x\\to a}{\\mathrm{lim}}g(x)[\/latex]<\/td>\n<\/tr>\n \n Limit of a quotient<\/strong><\/td>\n [latex]\\underset{x\\to a}{\\mathrm{lim}}\\frac{f(x)}{g(x)}=\\frac{\\underset{x\\to a}{\\mathrm{lim}}f(x)}{\\underset{x\\to a}{\\mathrm{lim}}g(x)}[\/latex], where [latex]\\underset{x\\to a}{\\mathrm{lim}}g(x)\\neq 0[\/latex]<\/td>\n<\/tr>\n \n Limit of a power<\/strong><\/td>\n [latex]\\underset{x\\to a}{\\mathrm{lim}}[f(x)]^{n}=[\\underset{x\\to a}{\\mathrm{lim}}f(x)]^{n}[\/latex]<\/td>\n<\/tr>\n \n Limit of a root<\/strong><\/td>\n [latex]\\underset{x\\to a}{\\mathrm{lim}}\\sqrt[n]{f(x)}=\\sqrt[n]{\\underset{x\\to a}{\\mathrm{lim}}f(x)}[\/latex]<\/td>\n<\/tr>\n \n Limit of a polynomial<\/strong><\/td>\n [latex]\\underset{x\\to a}{\\mathrm{lim}}p(x)=p(a)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n Glossary<\/h2>\n