91Ó°ÊÓ

Find the limits \(\lim \left(\frac{1}{t^{3 / 5}}+7\right)\) as a. \(t \rightarrow 0^{+}\) b. \(t \rightarrow 0^{-}\)

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form $$\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$ occur frequently in calculus. In Exercises \(57-62,\) evaluate this limit for the given value of \(x\) and function \(f\) $$f(x)=1 / x, \quad x=-2$$

A function discontinuous at every point a. Use the fact that every nonempty interval of real numbers contains both rational and irrational numbers to show that the function $$ f(x)=\left\\{\begin{array}{ll} 1, & \text { if } x \text { is rational } \\ 0, & \text { if } x \text { is irrational } \end{array}\right. $$ is discontinuous at every point. b. Is \(f\) right-continuous or left-continuous at any point?

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form $$\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$ occur frequently in calculus. In Exercises \(57-62,\) evaluate this limit for the given value of \(x\) and function \(f\) $$f(x)=\sqrt{x}, \quad x=7$$

Find the limits \(\lim \left(\frac{1}{x^{2 / 3}}+\frac{2}{(x-1)^{2 / 3}}\right)\) as a. \(x \rightarrow 0^{+}\) b. \(x \rightarrow 0\) c. \(x \rightarrow 1^{+}\) d. \(x \rightarrow 1^{-}\)

Find the limits \(\lim \left(\frac{1}{x^{1 / 3}}-\frac{1}{(x-1)^{4 / 3}}\right)\) as a. \(x \rightarrow 0^{+}\) b. \(x \rightarrow 0\) c. \(x \rightarrow 1^{+}\) d. \(x \rightarrow 1^{-}\)

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form $$\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$ occur frequently in calculus. In Exercises \(57-62,\) evaluate this limit for the given value of \(x\) and function \(f\) $$f(x)=\sqrt{3 x+1}, \quad x=0$$

If functions \(f(x)\) and \(g(x)\) are continuous for \(0 \leq x \leq 1,\) could \(f(x) / g(x)\) possibly be discontinuous at a point of [0,1]\(?\) Give reasons for your answer.

Graph the rational functions. Include the graphs and equations of the asymptotes and dominant terms. $$y=\frac{1}{x-1}$$

$$\begin{aligned} &\text { If } \sqrt{5-2 x^{2}} \leq f(x) \leq \sqrt{5-x^{2}} \text { for }-1 \leq x \leq 1, \text { find }\\\ &\lim _{x \rightarrow 0} f(x) \end{aligned}$$