Not all functions or their limits involve simple addition, subtraction, or multiplication. Some may include polynomials. Recall that a polynomial is an expression consisting of the sum of two or more terms, each of which consists of a constant and a variable raised to a nonnegative integral power. To find the limit of a polynomial function, we can find the limits of the individual terms of the function, and then add them together. Also, the limit of a polynomial function as [latex]x[/latex] approaches [latex]a[/latex] is equivalent to simply evaluating the function for [latex]a[/latex].
How To: Given a function containing a polynomial, find its limit.
Use the properties of limits to break up the polynomial into individual terms.
Find the limits of the individual terms.
Add the limits together.
Alternatively, evaluate the function for [latex]a[/latex].
[latex]\begin{align}\underset{x\to 3}{\mathrm{lim}}\left(5{x}^{2}\right)&=5\underset{x\to 3}{\mathrm{lim}}\left({x}^{2}\right)&& \text{Constant times a function property} \\ &=5\left({3}^{2}\right)&& \text{Function raised to an exponent property} \\ &=45 \end{align}[/latex]
[latex]\begin{align}\underset{x\to 5}{\mathrm{lim}}\left(2{x}^{3}-3x+1\right)&=\underset{x\to 5}{\mathrm{lim}}\left(2{x}^{3}\right)-\underset{x\to 5}{\mathrm{lim}}\left(3x\right)+\underset{x\to 5}{\mathrm{lim}}\left(1\right)&& \text{Sum of functions} \\ &=\underset{x\to 5}{2\mathrm{lim}}\left({x}^{3}\right)-\underset{x\to 5}{3\mathrm{lim}}\left(x\right)+\underset{x\to 5}{\mathrm{lim}}\left(1\right)&& \text{Constant times a function} \\ &=2\left({5}^{3}\right)-3\left(5\right)+1&& \text{Function raised to an exponent} \\ &=236&& \text{Evaluate} \end{align}[/latex]
Evaluate the following limit: [latex]\underset{x\to -1}{\mathrm{lim}}\left({x}^{4}-4{x}^{3}+5\right)[/latex].
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Finding the Limit of a Power or a Root
When a limit includes a power or a root, we need another property to help us evaluate it. The square of the limit of a function equals the limit of the square of the function; the same goes for higher powers. Likewise, the square root of the limit of a function equals the limit of the square root of the function; the same holds true for higher roots.
Evaluate the following limit: [latex]\underset{x\to -4}{\mathrm{lim}}{\left(10x+36\right)}^{3}[/latex].
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Some functions may be algebraically rearranged so that one can evaluate the limit of a simplified equivalent form of the function. For example in [latex]\underset{x\to 2}{\mathrm{lim}}\left(\frac{{x}^{2}+6x+8}{x - 2}\right)[/latex]