Home > APCALC > Chapter 4 > Lesson 4.4.2 > Problem4-148

4-148.

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.

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

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 $y = 0$.

3. $\lim\limits _ { x \rightarrow \pi } \frac { \operatorname { cos } ( x ) + 1 } { x - \pi }$

This is Ana's Definition of the Derivative:

$\lim \limits_{x\rightarrow a }\frac{f(x)-f(a)}{x-a}$

$f\left(x\right)=\cos x$ and $a = π$

So Ana's derivative can be evaluated as:

$f'(\pi )=\lim \limits_{x\rightarrow \pi }\frac{\text{cox}x-\text{cos}\pi }{x-\pi }.$

The limit is equal to $f^\prime \left(π\right)$.