- Find the limit of a sum, a difference, and a product.
- Find the limit of a polynomial.
- Find the limit of a power or a root.
- Find the limit of a quotient.
Finding the Limit of a Sum, a Difference, and a Product
Graphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the properties of limits, which is a collection of theorems for finding limits.
Knowing the properties of limits allows us to compute limits directly. We can add, subtract, multiply, and divide the limits of functions as if we were performing the operations on the functions themselves to find the limit of the result. Similarly, we can find the limit of a function raised to a power by raising the limit to that power. We can also find the limit of the root of a function by taking the root of the limit. Using these operations on limits, we can find the limits of more complex functions by finding the limits of their simpler component functions.
Properties of Limits
Let [latex]a,k,A[/latex], and [latex]B[/latex] represent real numbers, and [latex]f[/latex] and [latex]g[/latex] be functions, such that [latex]\underset{x\to a}{\mathrm{lim}}f\left(x\right)=A[/latex] and [latex]\underset{x\to a}{\mathrm{lim}}g\left(x\right)=B[/latex]. For limits that exist and are finite, the properties of limits are summarized in the table below.
| Constant, k | [latex]\underset{x\to a}{\mathrm{lim}}k=k[/latex] |
| Constant times a function | [latex]\underset{x\to a}{\mathrm{lim}}\left[k\cdot f\left(x\right)\right]=k\underset{x\to a}{\mathrm{lim}}f\left(x\right)=kA[/latex] |
| Sum of functions | [latex]\underset{x\to a}{\mathrm{lim}}\left[f\left(x\right)+g\left(x\right)\right]=\underset{x\to a}{\mathrm{lim}}f\left(x\right)+\underset{x\to a}{\mathrm{lim}}g\left(x\right)=A+B[/latex] |
| Difference of functions | [latex]\underset{x\to a}{\mathrm{lim}}\left[f\left(x\right)-g\left(x\right)\right]=\underset{x\to a}{\mathrm{lim}}f\left(x\right)-\underset{x\to a}{\mathrm{lim}}g\left(x\right)=A-B[/latex] |
| Product of functions | [latex]\underset{x\to a}{\mathrm{lim}}\left[f\left(x\right)\cdot g\left(x\right)\right]=\underset{x\to a}{\mathrm{lim}}f\left(x\right)\cdot \underset{x\to a}{\mathrm{lim}}g\left(x\right)=A\cdot B[/latex] |
| Quotient of functions | [latex]\underset{x\to a}{\mathrm{lim}}\dfrac{f\left(x\right)}{g\left(x\right)}=\dfrac{\underset{x\to a}{\mathrm{lim}}f\left(x\right)}{\underset{x\to a}{\mathrm{lim}}g\left(x\right)}=\dfrac{A}{B},B\ne 0[/latex] |
| Function raised to an exponent | [latex]\underset{x\to a}{\mathrm{lim}}{\left[f\left(x\right)\right]}^{n}={\left[\underset{x\to \infty }{\mathrm{lim}}f\left(x\right)\right]}^{n}={A}^{n}[/latex], where [latex]n[/latex] is a positive integer |
| nth root of a function, where n is a positive integer | [latex]\underset{x\to a}{\mathrm{lim}}\sqrt[n]{f\left(x\right)}=\sqrt[n]{\underset{x\to a}{\mathrm{lim}}\left[f\left(x\right)\right]}=\sqrt[n]{A}[/latex] |
| Polynomial function | [latex]\underset{x\to a}{\mathrm{lim}}p\left(x\right)=p\left(a\right)[/latex] |