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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 approaches is equivalent to simply evaluating the function for .
Given a function containing a polynomial, find its limit.
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
We will take the limit of the function as approaches 2 and raise the result to the 5 th power.
If we can’t directly apply the properties of a limit, for example in , can we still determine the limit of the function as approaches ?
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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