{"id":2847,"date":"2025-08-13T20:14:02","date_gmt":"2025-08-13T20:14:02","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2847"},"modified":"2025-12-02T22:48:34","modified_gmt":"2025-12-02T22:48:34","slug":"finding-limits-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/finding-limits-learn-it-2\/","title":{"raw":"Finding Limits: Learn It 2","rendered":"Finding Limits: Learn It 2"},"content":{"raw":"

Understanding Left-Hand Limits and Right-Hand Limits<\/h2>\r\nWe can approach the input of a function from either side of a value\u2014from the left or the right. The table below shows the values of\r\n
[latex]f\\left(x\\right)=x+1,x\\ne 7[\/latex]<\/div>\r\nas described earlier.\r\n\r\n\"Table\r\n\r\nValues described as \"from the left\" are less than the input value 7 and would therefore appear to the left of the value on a number line. The input values that approach 7 from the left are [latex]6.9[\/latex], [latex]6.99[\/latex], and [latex]6.999[\/latex]. The corresponding outputs are [latex]7.9,7.99[\/latex], and [latex]7.999[\/latex]. These values are getting closer to 8. The limit of values of [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches from the left is known as the left-hand limit. For this function, 8 is the left-hand limit of the function [latex]f\\left(x\\right)=x+1,x\\ne 7[\/latex] as [latex]x[\/latex] approaches 7.\r\n\r\nValues described as \"from the right\" are greater than the input value 7 and would therefore appear to the right of the value on a number line. The input values that approach 7 from the right are [latex]7.1[\/latex], [latex]7.01[\/latex], and [latex]7.001[\/latex]. The corresponding outputs are [latex]8.1[\/latex], [latex]8.01[\/latex], and [latex]8.001[\/latex]. These values are getting closer to 8. The limit of values of [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches from the right is known as the right-hand limit. For this function, 8 is also the right-hand limit of the function [latex]f\\left(x\\right)=x+1,x\\ne 7[\/latex] as [latex]x[\/latex] approaches 7.\r\n\r\nWe can get the output of the function within a distance of 0.1 from 8 by using an input within a distance of 0.1 from 7. In other words, we need an input [latex]x[\/latex] within the interval [latex]6.9<x<7.1[\/latex] to produce an output value of [latex]f\\left(x\\right)[\/latex] within the interval [latex]7.9<f\\left(x\\right)<8.1[\/latex].\r\n\r\nWe also see that we can get output values of [latex]f\\left(x\\right)[\/latex] successively closer to 8 by selecting input values closer to 7. In fact, we can obtain output values within any specified interval if we choose appropriate input values.\r\n\r\nFrom the graph of [latex]f\\left(x\\right)[\/latex], we observe the output can get infinitesimally close to [latex]L=8[\/latex] as [latex]x[\/latex] approaches 7 from the left and as [latex]x[\/latex] approaches 7 from the right.\r\n\r\nTo indicate the left-hand limit, we write\r\n
[latex]\\underset{x\\to {7}^{-}}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex].<\/div>\r\nTo indicate the right-hand limit, we write\r\n
[latex]\\underset{x\\to {7}^{+}}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex].<\/div>\r\n
\r\n\r\n[caption id=\"attachment_4973\" align=\"aligncenter\" width=\"467\"]\"Graph The left- and right-hand limits are the same for this function.[\/caption]\r\n\r\n<\/div>\r\n
<\/div>\r\n
\r\n

one-sided limits<\/h3>\r\nThe left-hand limit<\/strong> of a function [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] from the left is equal to [latex]L[\/latex], denoted by\r\n

[latex]\\underset{x\\to {a}^{-}}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex].<\/p>\r\nThe values of [latex]f\\left(x\\right)[\/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\\ne a[\/latex].\r\n\r\nThe right-hand limit<\/strong> of a function [latex]f\\left(x\\right)[\/latex], as [latex]x[\/latex] approaches [latex]a[\/latex] from the right, is equal to [latex]L[\/latex], denoted by\r\n

[latex]\\underset{x\\to {a}^{+}}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex].<\/p>\r\nThe values of [latex]f\\left(x\\right)[\/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] but greater than [latex]a[\/latex]. Both [latex]a[\/latex] and [latex]L[\/latex] are real numbers.\r\n\r\n<\/section>\r\n

Understanding Two-Sided Limits<\/h2>\r\nIn the previous example, the left-hand limit and right-hand limit as [latex]x[\/latex] approaches [latex]a[\/latex] are equal. If the left- and right-hand limits are equal, we say that the function [latex]f\\left(x\\right)[\/latex] has a two-sided limit<\/strong> as [latex]x[\/latex] approaches [latex]a[\/latex]. More commonly, we simply refer to a two-sided limit as a limit. If the left-hand limit does not equal the right-hand limit, or if one of them does not exist, we say the limit does not exist.\r\n\r\n
\r\n

two-sided limits<\/h3>\r\nThe limit of a function [latex]f\\left(x\\right)[\/latex], as [latex]x[\/latex] approaches [latex]a[\/latex], is equal to [latex]L[\/latex], that is,\r\n

[latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex]<\/p>\r\nif and only if\r\n

[latex]\\underset{x\\to {a}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {a}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex].<\/p>\r\nIn other words, the left-hand limit of a function [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] is equal to the right-hand limit of the same function as [latex]x[\/latex] approaches [latex]a[\/latex]. If such a limit exists, we refer to the limit as a two-sided limit. Otherwise we say the limit does not exist.\r\n\r\n<\/section>","rendered":"

Understanding Left-Hand Limits and Right-Hand Limits<\/h2>\n

We can approach the input of a function from either side of a value\u2014from the left or the right. The table below shows the values of<\/p>\n

[latex]f\\left(x\\right)=x+1,x\\ne 7[\/latex]<\/div>\n

as described earlier.<\/p>\n

\"Table<\/p>\n

Values described as “from the left” are less than the input value 7 and would therefore appear to the left of the value on a number line. The input values that approach 7 from the left are [latex]6.9[\/latex], [latex]6.99[\/latex], and [latex]6.999[\/latex]. The corresponding outputs are [latex]7.9,7.99[\/latex], and [latex]7.999[\/latex]. These values are getting closer to 8. The limit of values of [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches from the left is known as the left-hand limit. For this function, 8 is the left-hand limit of the function [latex]f\\left(x\\right)=x+1,x\\ne 7[\/latex] as [latex]x[\/latex] approaches 7.<\/p>\n

Values described as “from the right” are greater than the input value 7 and would therefore appear to the right of the value on a number line. The input values that approach 7 from the right are [latex]7.1[\/latex], [latex]7.01[\/latex], and [latex]7.001[\/latex]. The corresponding outputs are [latex]8.1[\/latex], [latex]8.01[\/latex], and [latex]8.001[\/latex]. These values are getting closer to 8. The limit of values of [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches from the right is known as the right-hand limit. For this function, 8 is also the right-hand limit of the function [latex]f\\left(x\\right)=x+1,x\\ne 7[\/latex] as [latex]x[\/latex] approaches 7.<\/p>\n

We can get the output of the function within a distance of 0.1 from 8 by using an input within a distance of 0.1 from 7. In other words, we need an input [latex]x[\/latex] within the interval [latex]6.9[latex]\\underset{x\\to {7}^{-}}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex].<\/div>\n

To indicate the right-hand limit, we write<\/p>\n

[latex]\\underset{x\\to {7}^{+}}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex].<\/div>\n
\n
\"Graph
The left- and right-hand limits are the same for this function.<\/figcaption><\/figure>\n<\/div>\n
<\/div>\n
\n

one-sided limits<\/h3>\n

The left-hand limit<\/strong> of a function [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] from the left is equal to [latex]L[\/latex], denoted by<\/p>\n

[latex]\\underset{x\\to {a}^{-}}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex].<\/p>\n

The values of [latex]f\\left(x\\right)[\/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]xright-hand limit<\/strong> of a function [latex]f\\left(x\\right)[\/latex], as [latex]x[\/latex] approaches [latex]a[\/latex] from the right, is equal to [latex]L[\/latex], denoted by<\/p>\n

[latex]\\underset{x\\to {a}^{+}}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex].<\/p>\n

The values of [latex]f\\left(x\\right)[\/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] but greater than [latex]a[\/latex]. Both [latex]a[\/latex] and [latex]L[\/latex] are real numbers.<\/p>\n<\/section>\n

Understanding Two-Sided Limits<\/h2>\n

In the previous example, the left-hand limit and right-hand limit as [latex]x[\/latex] approaches [latex]a[\/latex] are equal. If the left- and right-hand limits are equal, we say that the function [latex]f\\left(x\\right)[\/latex] has a two-sided limit<\/strong> as [latex]x[\/latex] approaches [latex]a[\/latex]. More commonly, we simply refer to a two-sided limit as a limit. If the left-hand limit does not equal the right-hand limit, or if one of them does not exist, we say the limit does not exist.<\/p>\n

\n

two-sided limits<\/h3>\n

The limit of a function [latex]f\\left(x\\right)[\/latex], as [latex]x[\/latex] approaches [latex]a[\/latex], is equal to [latex]L[\/latex], that is,<\/p>\n

[latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex]<\/p>\n

if and only if<\/p>\n

[latex]\\underset{x\\to {a}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {a}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex].<\/p>\n

In other words, the left-hand limit of a function [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] is equal to the right-hand limit of the same function as [latex]x[\/latex] approaches [latex]a[\/latex]. If such a limit exists, we refer to the limit as a two-sided limit. Otherwise we say the limit does not exist.<\/p>\n<\/section>\n","protected":false},"author":13,"menu_order":7,"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":"- Select Header -","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\/2847"}],"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\/13"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2847\/revisions"}],"predecessor-version":[{"id":4974,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2847\/revisions\/4974"}],"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\/2847\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2847"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2847"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2847"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2847"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}