91影视

Find each of the following limits.

$\begin{array}{l}\underset{h鈫�0}{\lim }\frac{(1-h)-1}{h}\\ \underset{h鈫�0}{\lim }\frac{\left(3(-1+h{)}^2+1\right)-4}{h}\\ \underset{h鈫�0}{\lim }\frac{\frac{1}{2+h}-0.5}{h}\\ \underset{z鈫�2}{\lim }\frac{z^2-4}{z-2}\\ \underset{z鈫�4}{\lim }\frac{(1-3z)+11}{z-4}\\ \underset{z鈫�1}{\lim }\frac{\frac{1}{z}-1}{z-1}\end{array}$

Instead of choosing small values of h, we could have chosen values of z close to c. What limit involving z instead of h is equivalent to the one involving h?

Show that each of the following limits is of the form $\frac{0}{0}$and then use L鈥橦opital鈥檚 rule to calculate the limit: 藛

Express the constant multiple, sum, and difference rules in Leibniz/operator notation

Finding critical points: For each of the following functions $f$, find all of the $x$-values for which $f\text{'}\left(x\right)=0$and all of the $x$-values for which $f\text{'}\left(x\right)$does not exist.

role="math" localid="1649086262895" $f\left(x\right)=\frac{\sqrt{x}}{x\sqrt{x-1}}$

Use the preceding two derivative formulas to make a conjecture about a formula for $\frac{d}{dx}\left(x^n\right)$, whererole="math" $n$ is a positive integer.

Basic definition-of-derivative calculations: Find the derivatives of the functions that follow, using (a) the $h鈫�0$definition of the derivative and (b) the $z鈫�x$definition of the derivative.

$f\left(x\right)=\frac{1}{x}$

Find the derivatives of$f\left(x\right)=\frac{1}{x}.$

Translate expressions written in Leibniz notation to 鈥減rime鈥� notation, and vice versa.

$f^{\text{'}\text{'}}\left(x\right)$

Explain why it is not a simple task to calculate the slope of the tangent line to a function $f$at a point $x=c$. Shouldn鈥檛 calculating the slope of a line be really easy? What goes wrong here?