91Ó°ÊÓ

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{t \rightarrow-2}(t+1)^{9}\left(t^{2}-1\right)$$

Find the area of the region that lies under the graph of \(f\) over the given interval. $$f(x)=3 x^{2}, \quad 0 \leq x \leq 2$$

Estimating Limits Numerically and Graphically Use a table of values to estimate the value of the limit. Then use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 1}\left(\frac{1}{\ln x}-\frac{1}{x-1}\right)$$

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow 1}\left(\frac{x^{4}+x^{2}-6}{x^{4}+2 x+3}\right)^{2}$$

Find an equation of the tangent line to the curve at the given point. Graph the curve and the tangent line. $$y=\frac{x}{x-1}, \quad \text { at }(2,2)$$

Find the limit. $$\lim _{x \rightarrow-\infty}\left(\frac{x-1}{x+1}+6\right)$$

Find the limit. $$\lim _{x \rightarrow-\infty}\left(\frac{3-x}{3+x}-2\right)$$

Find the area of the region that lies under the graph of \(f\) over the given interval. $$f(x)=x+x^{2}, \quad 0 \leq x \leq 1$$

Estimating Limits Numerically and Graphically Use a table of values to estimate the value of the limit. Then use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 0} \frac{\tan 2 x}{\tan 3 x}$$

Find an equation of the tangent line to the curve at the given point. Graph the curve and the tangent line. $$y=\frac{1}{x^{2}}, \quad \text { at }(-1,1)$$