Exam 12: Limits, Derivatives, and Definite Integrals

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow 0} \frac{x^{2}-4}{x^{3}+8}$

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow \frac{1}{3}} \frac{6 x^{2}+x-1}{3 x-1}$

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow 2} \frac{x^{4}-2 x^{3}}{x-2}$

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow 4^{-}} f(x)$

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow \infty} \frac{6 x^{2}+5}{2 x^{2}-4 x+7}$

Find the slope of the tangent line to the graph of $f(x)=4-3 x^{2}$ at the point where $x=-2$ .

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow-5} \frac{x^{2}+7 x+10}{x+5}$

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow 5^{-}} f(x)$ where $f(x)= \begin{cases}x^{2}+3 x & \text { if } x \leq 5 \\ 4 x+15 & \text { if } x>5\end{cases}$

Use a calculator to estimate $f^{\prime}(1)$ , where $f(x)=\frac{2-3 e^{x}}{3 x}$ .

Use a calculator to estimate $f^{\prime}(1)$ , where $f(x)=\frac{2+3 e^{-x}}{2 x}$ .

Find the equation of the tangent line to the graph of $f(x)=\frac{10}{x}$ at the point $(-2,-5)$ .

Approximate the area under the graph of $f(x)=-x^{2}+9$ from $x=-1$ to $x=3$ and above the $x$ -axis, using four rectangles. Let the height of each rectangle be the function value at the midpoint of the interval.

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow 4} \frac{x^{2}-10 x+24}{x-4}$

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow \infty} \frac{3 x^{3}+7}{x^{4}+3 x^{2}+1}$

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow 3} \sqrt{16+x^{2}}$

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow 9} \sqrt{x^{2}-17}$

find the value of the given limit if it exists. If the function approaches $\infty$ or $-\infty$ , say so. - $\lim _{x \rightarrow 0} \frac{x^{3}+2 x+3}{x^{3}-7}$