### Home > CALC > Chapter 4 > Lesson 4.4.2 > Problem4-137

4-137.

Without your calculator, evaluate the following limits.

1. $\lim\limits_ { x \rightarrow 3 ^ { + } } \sqrt { x - 3 }$

Notice that this is a right-sided limit because $x = 3$ is an endpoint on $y=\sqrt{x-3}.$
There is no limit from the left of the endpoint; so, consequently, this must be written as a one-sided limit.

1. $\lim\limits_ { x \rightarrow \infty } \frac { x ^ { 2 } - 2 x + 1 } { x ^ { 3 } }$

This is a limit to infinity. We are looking for end-behavior. Compare the highest-power term in the numerator and the denominator.

$\lim\limits_{x\rightarrow \infty }\frac{x^{2}}{x^{3}}=\lim\limits_{x\rightarrow \infty }\frac{1}{x}= 0$
There is a horizontal asymptote of $y = 0$.

1. $\lim\limits_ { x \rightarrow \pi } \frac { \operatorname { cos } x + 1 } { x - \pi }$

This is Ana's Definition of the Derivative:

f$(x) =\operatorname{cos}x$ and $a = π$
So Ana's derivative can be evaluated as:
The limit is equal to $f^\prime(π)$.
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