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Evaluate $\text{\hspace{0.17em}}\underset{x\to 3}{\mathrm{lim}}\left(2x+5\right).$
Evaluate the following limit: $\text{\hspace{0.17em}}\underset{x\to -12}{\mathrm{lim}}\left(-2x+2\right).$
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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 $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is equivalent to simply evaluating the function for $\text{\hspace{0.17em}}a$ .
Given a function containing a polynomial, find its limit.
Evaluate $\text{\hspace{0.17em}}\underset{x\to 3}{\mathrm{lim}}\left(5{x}^{2}\right).$
Evaluate $\text{\hspace{0.17em}}\underset{x\to 4}{\mathrm{lim}}({x}^{3}-5).$
59
Evaluate $\text{\hspace{0.17em}}\underset{x\to 5}{\mathrm{lim}}\left(2{x}^{3}-3x+1\right).$
Evaluate the following limit: $\text{\hspace{0.17em}}\underset{x\to -1}{\mathrm{lim}}\left({x}^{4}-4{x}^{3}+5\right).$
10
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 $\text{\hspace{0.17em}}\underset{x\to 2}{\mathrm{lim}}{\left(3x+1\right)}^{5}.$
We will take the limit of the function as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches 2 and raise the result to the 5 ^{th} power.
Evaluate the following limit: $\underset{x\to -4}{\mathrm{lim}}{\left(10x+36\right)}^{3}.$
$-64$
If we can’t directly apply the properties of a limit, for example in $\underset{x\to 2}{\mathrm{lim}}(\frac{{x}^{2}+6x+8}{x-2})$ , can we still determine the limit of the function as $x$ approaches $a$ ?
Yes. Some functions may be algebraically rearranged so that one can evaluate the limit of a simplified equivalent form of the function.
Finding the limit of a function expressed as a quotient can be more complicated. We often need to rewrite the function algebraically before applying the properties of a limit. If the denominator evaluates to 0 when we apply the properties of a limit directly, we must rewrite the quotient in a different form. One approach is to write the quotient in factored form and simplify.
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