91Ó°ÊÓ

Find the slope of the tangent line to the graph of \(f\) at the given point. $$f(x)=3 x+4 \quad \text { at }(1,7)$$

Suppose that \(\lim _{x \rightarrow a} f(x)=-3 \quad \lim _{x \rightarrow a} g(x)=0 \quad \lim _{x \rightarrow a} h(x)=8\) Find the value of the given limit. If the limit does not exist, explain why. (a) \(\lim _{x \rightarrow a}[f(x)+h(x)]\) (b) \(\lim _{x \rightarrow a}[f(x)]^{2}\) (c) \(\lim _{x \rightarrow a} \sqrt[3]{h(x)}\) (d) \(\lim _{x \rightarrow a} \frac{1}{f(x)}\) (e) \(\lim _{x \rightarrow a} \frac{f(x)}{h(x)}\) (f) \(\lim _{x \rightarrow a} \frac{g(x)}{f(x)}\) (g) \(\lim _{x \rightarrow a} \frac{f(x)}{g(x)}\) (h) \(\lim _{x \rightarrow a} \frac{2 f(x)}{h(x)-f(x)}\)

Complete the table of values (to five decimal places) and use the table to estimate the value of the limit. $$\lim _{x \rightarrow 4} \frac{\sqrt{x}-2}{x-4}$$ $$\begin{array}{|c|c|c|c||c|c|c|} \hline x & 3.9 & 3.99 & 3.999 & 4.001 & 4.01 & 4.1 \\ \hline f(x) & & & & & & \\ \hline \end{array}$$

Find the slope of the tangent line to the graph of \(f\) at the given point. $$f(x)=5-2 x \text { at }(-3,11)$$

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow 4}\left(5 x^{2}-2 x+3\right)$$

Find the slope of the tangent line to the graph of \(f\) at the given point. $$f(x)=4 x^{2}-3 x \quad \text { at }(-1,7)$$

Find the limit. $$\lim _{x \rightarrow \infty} \frac{6}{x}$$

Complete the table of values (to five decimal places) and use the table to estimate the value of the limit. $$\lim _{x \rightarrow 1} \frac{x-1}{x^{3}-1}$$ $$\begin{array}{|c|c|c|c||c|c|c|} \hline x & 0.9 & 0.99 & 0.999 & 1.001 & 1.01 & 1.1 \\ \hline f(x) & & & & & & \\ \hline \end{array}$$

Find the limit. $$\lim _{x \rightarrow \infty} \frac{3}{x^{4}}$$

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow 3}\left(x^{3}+2\right)\left(x^{2}-5 x\right)$$