21–30. Derivatives

a. Use limits to find the derivative function f' for the following functions f.

f(x) = 5x+2; a=1, 2

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

f(−1)

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→−1^− f(x)

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→−1^+ f(x)

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→−1 f(x)

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

f(1)

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→1 f(x)

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→3^− f(x)

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→3^+ f(x)

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→3 f(x)

Determine the following limits at infinity.

lim x→ ∞ x^−6

Use the graph of $f$ in the figure to find the following values or state that they do not exist. <IMAGE>

$f\left(1\right)$

Use the graph of $f$ in the figure to find the following values or state that they do not exist. <IMAGE>

$f\left(0\right)$

Determine the following limits at infinity.

lim x→−∞ x^−11

Estimate lim θ→0 sin 2θ / sin θ using the graph in part (a).

Assume lim x→1 f(x)=8,lim x→1 g(x)=3, and lim x→1 h(x)=2 Compute the following limits and state the limit laws used to justify your computations.

lim x→1 (4f(x))

Let $f\left(x\right)=\frac{x^2-4}{x-2}$ . <IMAGE>

Calculate $f\left(x\right)$ for each value of $x$ in the following table.

Let $f\left(x\right)=\frac{x^2-4}{x-2}$. <IMAGE>

Make a conjecture about the value of ${\displaystyle\lim_{x\to2}\frac{x^2-4}{x-2}}$.

Suppose the rental cost for a snowboard is $25 for the first day (or any part of the first day) plus $15 for each additional day (or any part of a day).

Evaluate lim t→2.9 f(t).

Assume lim x→1 f(x)=8,lim x→1 g(x)=3, and lim x→1 h(x)=2 Compute the following limits and state the limit laws used to justify your computations.

lim x→1 (f(x)−g(x))

Let $g\left(t\right)=\frac{t-9}{\sqrt{t}-3}$.

Make two tables, one showing values of $g$ for $t=8.9,8.99$, and $8.999$ and one showing values of $g$ for $t=9.1,9.01$, and $9.001$.

Let $g\left(t\right)=\frac{t-9}{\sqrt{t}-3}$.

Make a conjecture about the value of ${\displaystyle\lim_{t\to9}\frac{t-9}{\sqrt{t}-3}}$.

Determine the following limits.​

lim x→1000 18π^2

Assume lim x→1 f(x)=8,lim x→1 g(x)=3, and lim x→1 h(x)=2 Compute the following limits and state the limit laws used to justify your computations.

lim x→1 f(x) / g(x)−h(x)

Determine the following limits.​

lim x→1 √5x+6

Determine the following limits.​

lim h→0 √5x + 5h − √5x / h, where x>0 Is constant

Given the function $f\left(x\right)=-16x^2+64x$, complete the following. <IMAGE>

Find the slopes of the secant lines that pass though the points $\left(x,f\left(x\right)\right)$ and $\left(2,f\left(2\right)\right)$, for $x=1.5,1.9,1.99,1.999,$ and $1.9999$ (see figure).

Given the function $f\left(x\right)=-16x^2+64x$, complete the following. <IMAGE>

Make a conjecture about the value of the limit of the slopes of the secant lines that pass through $\left(x,f\left(x\right)\right)$ and $\left(2,f\left(2\right)\right)$ as $x$ approaches $2$.

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.

c. lim x→0^− f(x)

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.

d. lim x→0^+ f(x)