91Ó°ÊÓ

Determine the following limits. $$\lim _{x \rightarrow-\infty}(x+\sqrt{x^{2}-5 x})$$

Evaluate limit and justify your answer. $$\lim _{t \rightarrow 4} \frac{t-4}{\sqrt{t}-2}$$

The Heaviside function The Heaviside function is used in engineering applications to model flipping a switch. It is defined as $$H(x)=\left\\{\begin{array}{ll}0 & \text { if } x<0 \\\1 & \text { if } x \geq0\end{array}\right.$$ a. Sketch a graph of \(H\) on the interval [-1,1] b. Does \(\lim _{\vec{x} \rightarrow 0} H(x)\) exist?

Find the following limits or state that they do not exist. Assume \(a, b, c,\) and k are fixed real numbers. $$\lim _{x \rightarrow 3} \frac{x^{2}-2 x-3}{x-3}$$

Limit proofs Use the precise definition of a limit to prove the following limits. Specify a relationship between \(\varepsilon\) and \(\delta\) that guarantees the limit exists. $$\begin{aligned} &\lim _{x \rightarrow 4} \frac{x-4}{\sqrt{x}-2}=4(\text {Hint}:\text { Multiply the numerator and denomina- }\\\ &\text { tor by }\sqrt{x}+2 .) \end{aligned}$$

Evaluate limit and justify your answer. $$\lim _{x \rightarrow 1}\left(\frac{x+5}{x+2}\right)^{4}$$

Determine the following limits. $$\lim _{x \rightarrow \infty} \frac{\sin x}{e^{x}}$$

$$\lim _{x \rightarrow 1^{+}} \frac{x^{2}-5 x+6}{x-1}$$

Find the following limits or state that they do not exist. Assume \(a, b, c,\) and k are fixed real numbers. $$\lim _{x \rightarrow 4} \frac{x^{2}-16}{4-x}$$

Limit proofs Use the precise definition of a limit to prove the following limits. Specify a relationship between \(\varepsilon\) and \(\delta\) that guarantees the limit exists. \(\lim _{x \rightarrow 1 / 10} \frac{1}{x}=10\) (Hint: To find \(\delta,\) you need to bound \(x\) away from 0. So let \(\left.\left|x-\frac{1}{10}\right|<\frac{1}{20} .\right)\)