What mathematical constant does $\lim _ { x \rightarrow 0 } ( 1 + x ) ^ { 1 / x }$ appear to equal?

Describe a process for finding the slope of the line tangent to the graph of f at (a, f(a)).

Graph the function $f(x) = \frac{e^{-x}}{x{(x+2)}^2}$ using a graphing utility. ( Experiment with your choice of a graphing window.) Use your graph to discuss the following limits.

$$\lim _{x \rightarrow 0^{-}}f(x)$$

A projectile is fired vertically upward and has a position given by $s ( t ) = - 16 t ^ { 2 } + 128 t + 192$, for $0 \leq t \leq 9$. Graph the position function, for $0 \leq t \leq 9$. b. From the graph of the position function, identify the time at $t=a$

Graph the function $f(x) = \frac{e^{-x}}{x{(x+2)}^2}$ using a graphing utility. ( Experiment with your choice of a graphing window.) Use your graph to discuss the following limits.

$$\lim _{x \rightarrow -2}f(x)$$

Graph the function f(x)=

$$\frac { e ^ { - x } } { x ( x + 2 ) ^ { 2 } }$$

using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits. a.

$$\lim _ { x \rightarrow - 2 ^ { + } } f ( x )$$

, b.

$$\lim _ { x \rightarrow - 2 } f ( x )$$

, c.

$$\lim _ { x \rightarrow 0 } f ( x )$$

, d.

$$\lim _ { x \rightarrow 0 ^ { + } } f ( x )$$

Estimate the value of $\lim _ { x \rightarrow 0 } ( 1 + x ) ^ { 1 / x }$.

Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. $f ( x ) = \frac { 2 x ^ { 2 } + 3 x + 1 } { x ^ { 2 } + 5 x }$; a=5

Let us consider the following equation that is $f(x)=( 1 + x ) ^ { 1 / x }$. Make two tables, one showing values of f for x=0.01,0.001, 0.0001, and 0.00001 and one showing values of f for x=-0.01, -0.001, -00001, and -0.00001. Round your answers to five digits.

Graph the function $f(x) = \frac{e^{-x}}{x{(x+2)}^2}$ using a graphing utility. ( Experiment with your choice of a graphing window.) Use your graph to discuss the following limits.

$$\lim _{x \rightarrow -2^{+}}f(x)$$

Evaluate the following limits.

$$\lim _ { x \rightarrow \infty } \frac { \cos x ^ { 5 } } { \sqrt { x } }$$

Let us consider the equation which is given as $f(x)=x^3+3$ and note that $\lim _ { x \rightarrow 0 } f ( x ) = 3$. For each value of $\varepsilon$, use a graphing utility to find all values of $\delta \gt 0$ such that |f(x)-3|lt$\varepsilon$ whenever $0 \lt |x-0| \lt \delta$. Sketch graphs illustrating your work for $\varepsilon = 0.5$.

Determine the following limits analytically. $\lim _ { x \rightarrow 1 } \frac { 1 - x ^ { 2 } } { x ^ { 2 } - 8 x + 7 }$

Graph the following function which is given as $f ( x ) = \frac { 1 } { x ^ { 2 } - x }$ using a graphing utility with the window $[ - 1,2 ] \times [ - 10,10 ]$. Use your graph to determine the following limit which is $\lim _ { x \rightarrow 1 ^ { + } } f ( x )$