Numerical, Graphical, and Analytic Analysis In Exercises 33 and 34, consider(a) Complete

In Exercises 1 and 2, determine whether the problem can be solved using precalculus, or if calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, explain your reasoning. Use a graphical or numerical approach to estimate the solution. Find the distance between the points (1, 1) and (3, 9) along the curve \(y=x^{2}\). Equation Transcription: Text Transcription: y = x^2

In Exercises 1 and 2, determine whether the problem can be solved using precalculus, or if calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, explain your reasoning. Use a graphical or numerical approach to estimate the solution. Find the distance between the points (1, 1) and (3, 9) along the line y = 4x - 3.

In Exercises 36, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result.lim x0 4
x 2
2 x

In Exercises 36, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result.lim x0 4
x 2
2 x

In Exercises 36, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result.lim x0 20
ex2
1

In Exercises 36, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result.lim x0 ln
x 5
ln 5 x

In Exercises 7–10, use the graph to determine each limit. \(h(x)=\frac{x^{2}-2 x}{x}\) (a) \(\lim \limits_{x \rightarrow 0} h(x)\) (b) \(\lim \limits_{x \rightarrow-1} h(x)\) Equation Transcription: Text Transcription: h(x) = x^2 - 2x/x lim_x right arrow 0 h(x) lim_x right arrow -1 h(x)

In Exercises 7–10, use the graph to determine each limit. \(g(x)=\frac{3 x}{x-2}\) (a) \(\lim \limits_{x \rightarrow 2} g(x)\) (b) \(\lim \limits_{x \rightarrow 0} g(x)\) Equation Transcription: Text Transcription: g(x) = 3x/x - 2 lim_x right arrow 2 g(x) lim_x right arrow 0 g(x)

In Exercises 7–10, use the graph to determine each limit. \(f(t)=\frac{\ln (t+2)}{t}\) (a) \(\lim \limits_{t \rightarrow 0} f(t)\) (b) \(\lim \limits_{t \rightarrow-1} f(t)\) Equation Transcription: Text Transcription: f(t) = ln(t + 2)/t lim_t right arrow 0 f(t) lim_t right arrow -1 f(t)

In Exercises 7–10, use the graph to determine each limit. \(g(x)=e^{-x / 2} \sin \pi x\) (a) \(\lim \limits_{x \rightarrow 0} g(x)\) (b) \(\lim \limits_{x \rightarrow 2} g(x)\) Equation Transcription: Text Transcription: g(x) = e^-x/2 sin pi x lim_x right arrow 0 g(x) lim_x right arrow 2 g(x)

In Exercises 1114, find the limit Then use the - definition to prove that the limit is L.lim x x1
3
x

In Exercises 1114, find the limit Then use the - definition to prove that the limit is L.lim x9 lim x

In Exercises 1114, find the limit Then use the - definition to prove that the limit is L.lim 9 x2
x2
3

In Exercises 1114, find the limit Then use the - definition to prove that the limit is L.im x5 lim 9 x

In Exercises 1530, find the limit (if it exists).m t4 t 2

In Exercises 1530, find the limit (if it exists).lim y4 3 y
1 l

In Exercises 1530, find the limit (if it exists).lim t
2 t 2 t 2
4

In Exercises 1530, find the limit (if it exists).lim t3 t 2
9 t
3

In Exercises 15–30, find the limit (if it exists). \(\lim \limits_{x \rightarrow 4} \frac{\sqrt{x}-2}{x-4}\) Equation Transcription: Text Transcription: lim_x right arrow 4 sqrt x - 2/x - 4

In Exercises 15–30, find the limit (if it exists). \(\lim \limits_{x \rightarrow 0} \frac{\sqrt{4+x}-2}{x}\) Equation Transcription: Text Transcription: lim_x right arrow 0 sqrt 4 + x - 2/x

In Exercises 1530, find the limit (if it exists).lim x0 1
x 1
1 x

In Exercises 1530, find the limit (if it exists).lim s0
11 s
1 s

In Exercises 1530, find the limit (if it exists).lim x
5 x3 125 x 5 l

In Exercises 1530, find the limit (if it exists).lim x
2 x2
4 x3 8 l

In Exercises 15–30, find the limit (if it exists). \(\lim \limits_{x \rightarrow 0} \frac{1-\cos x}{\sin x}\) Equation Transcription: Text Transcription: lim_x right arrow 0 1 - cos x/sin x

In Exercises 15–30, find the limit (if it exists). \(\lim \limits_{x \rightarrow \pi / 4} \frac{4 x}{\tan x}\) Equation Transcription: Text Transcription: lim_x right arrow pi/4 4x/tan x

In Exercises 1530, find the limit (if it exists).im x0 sin
6 x

12 x

In Exercises 1530, find the limit (if it exists).im x0 cos
 x 1 x os
  cos  cos 
sin  sin  li

In Exercises 1530, find the limit (if it exists).lim x1 e x
1 sin x 2

In Exercises 1530, find the limit (if it exists).lim x2 ln
x
12 ln
x
1 l

In Exercises 31 and 32, evaluate the limit given thatlim . xc fx 3 4lim f
x 2g
x xc  f
xg
x limxc

In Exercises 31 and 32, evaluate the limit given thatm xc lim f
x 2g
x xc

Numerical, Graphical, and Analytic Analysis In Exercises 33 and 34, consider(a) Complete the table to estimate the limit. (b) Use a graphing utility to graph the function and use the graph to estimate the limit. (c) Rationalize the numerator to find the exact value of the limit analytically.f
x 2x 1
3 x
1 li

Numerical, Graphical, and Analytic Analysis In Exercises 33 and 34, consider(a) Complete the table to estimate the limit. (b) Use a graphing utility to graph the function and use the graph to estimate the limit. (c) Rationalize the numerator to find the exact value of the limit analytically.a3
b3
a
b
a2 ab b2  f
x 1

In Exercises 35 and 36, use the position function \(s(t)=-4.9 t^{2}+200\) which gives the height (in meters) of an object that has fallen from a height of 200 meters. The velocity at time t = a seconds is given by \(\lim \limits_{t \rightarrow a} \frac{s(a)-s(t)}{a-t}\). Find the velocity of the object when t = 4. Equation Transcription: Text Transcription: s(t) = -4.9t^2 + 200 lim_t right arrow a s(a) - s(t)/a - t

In Exercises 35 and 36, use the position function \(s(t)=-4.9 t^{2}+200\) which gives the height (in meters) of an object that has fallen from a height of 200 meters. The velocity at time t = a seconds is given by \(\lim \limits_{t \rightarrow a} \frac{s(a)-s(t)}{a-t}\). At what velocity will the object impact the ground? Equation Transcription: Text Transcription: s(t) = -4.9t^2 + 200 lim_t right arrow a s(a) - s(t)/a - t

In Exercises 37–42, find the limit (if it exists). If the limit does not exist, explain why. \(\lim \limits_{x \rightarrow 3^{-}} \frac{|x-3|}{x-3}\) Equation Transcription: Text Transcription: lim_x right arrow 3^- |x - 3|/x - 3

In Exercises 37–42, find the limit (if it exists). If the limit does not exist, explain why. \(\lim \limits_{x \rightarrow 4}[[x-1]]\) Equation Transcription: Text Transcription: lim_x right arrow 4 [[x - 1]]

In Exercises 37–42, find the limit (if it exists). If the limit does not exist, explain why. \(\lim \limits_{x \rightarrow 2} f(x)\), where \(f(x)=\left\{\begin{array}{ll} (x-2)^{2}, & x \leq 2 \\ 2-x, & x>2 \end{array}\right.\) Equation Transcription: { Text Transcription: lim_x right arrow 2 f(x) f(x) = {_2 - x, x > 2 ^(x - 2)^2, x leq 2

In Exercises 37–42, find the limit (if it exists). If the limit does not exist, explain why. \(\lim \limits_{x \rightarrow 1^{+}} g(x)\), where \(g(x)=\left\{\begin{array}{ll} \sqrt{1-x}, & x \leq 1 \\ x+1, & x>1 \end{array}\right.\) Equation Transcription: { Text Transcription: lim_x right arrow 1^+ g(x) g(x) = {_x + 1, x > 1 ^sqrt 1 - x, x leq 1

In Exercises 37–42, find the limit (if it exists). If the limit does not exist, explain why. \(\lim \limits_{t \rightarrow 1} h(t)\), where \(h(t)=\left\{\begin{array}{ll} t^{3}+1, & t<1 \\ \frac{1}{2}(t+1), & t \geq 1 \end{array}\right.\) Equation Transcription: { Text Transcription: lim_t right arrow 1 h(t) h(t) = {_1/2 (t + 1), t geq 1 ^t^3 + 1, t < 1

In Exercises 37–42, find the limit (if it exists). If the limit does not exist, explain why. \(\lim \limits_{s \rightarrow-2} f(s)\), where \(f(s)=\left\{\begin{array}{cc} -s^{2}-4 s-2, & s \leq-2 \\ s^{2}+4 s+6, & s>-2 \end{array}\right.\) Equation Transcription: { Text Transcription: lim_s right arrow -2 f(s) f(s) = {_s^2 + 4s + 6, s > -2 ^-s^2 - 4s - 2, s leq -2

In Exercises 4354, determine the intervals on which the function is continuous.f
x x 3 f

In Exercises 4354, determine the intervals on which the function is continuous.
x 3x2
x
2 x
1 f
x

In Exercises 4354, determine the intervals on which the function is continuous.
x  3x2
x
2, x
1 0, x  1 x 1 f
x

In Exercises 4354, determine the intervals on which the function is continuous.f
x  5
x, 2x
3, x 2 x > 2 f

In Exercises 4354, determine the intervals on which the function is continuous.f
x 1
x
22 f

In Exercises 4354, determine the intervals on which the function is continuous.f
x x 1 x f
x

In Exercises 4354, determine the intervals on which the function is continuous.
x 3 x 1 f
x

In Exercises 4354, determine the intervals on which the function is continuous.f
x x 1 2x 2 f
x

In Exercises 4354, determine the intervals on which the function is continuous.

In Exercises 4354, determine the intervals on which the function is continuous.f
x tan 2x x

In Exercises 4354, determine the intervals on which the function is continuous.g x
3
x 2e x4 f

In Exercises 4354, determine the intervals on which the function is continuous.h
x 5 ln g x
3
x

Determine the value of c such that the function is continuous on the entire real number line.f
x  x 3, cx 6, x 2 x > 2 h
x

Determine the values of b and c such that the function is continuous on the entire real number line.f
x  x 1, x2 bx c, 1 < x < 3 x
2 1 f
x

Use the Intermediate Value Theorem to show thatf
x 2x3
3 f

Delivery Charges The cost of sending an overnight package from New York to Atlanta is $9.80 for the first pound and $2.50 for each additional pound or fraction thereof. Use the greatest integer function to create a model for the cost of overnight delivery of a package weighing pounds. Use a graphing utility to graph the function and discuss its continuity.

Compound Interest A sum of $5000 is deposited in a savings plan that pays 12% interest compounded semiannually. The account balance after years is given by Use a graphing utility to graph the function and discuss its continuity.

Let (a) Find the domain of f. (b) Find (c) Findlim x1 f
x. l

In Exercises 6166, find the vertical asymptotes (if any) of the function.
x 1 2 x li

In Exercises 6166, find the vertical asymptotes (if any) of the function.h
x 4x 4
x2 g

In Exercises 6166, find the vertical asymptotes (if any) of the function.f
x f
x csc x 8

In Exercises 6166, find the vertical asymptotes (if any) of the function.
x
102 h
x

In Exercises 6166, find the vertical asymptotes (if any) of the function.g
x ln
9
x2 f

In Exercises 6166, find the vertical asymptotes (if any) of the function.f
x 10e
2x g

In Exercises 6778, find the one-sided limit.lim x
2
2x2 x 1 x 2 f

In Exercises 6778, find the one-sided limit.lim x
12 x 2x
1 lim

In Exercises 6778, find the one-sided limit.lim x
1 x 1 x3 1 lim

In Exercises 6778, find the one-sided limit.lim x
1
x 1 x4
1 l

In Exercises 6778, find the one-sided limit.lim x1
x2 2x 1 x
1 li

In Exercises 6778, find the one-sided limit.lim x
1 x2
2x 1 x 1 lim

In Exercises 6778, find the one-sided limit.im x0 sin 4x 5x

In Exercises 6778, find the one-sided limit.lim x0 sec x x l

In Exercises 6778, find the one-sided limit.im x0 csc 2x x

In Exercises 6778, find the one-sided limit.im x0
cos2 x x

In Exercises 6778, find the one-sided limit.lim x0 ln
sin x l

In Exercises 6778, find the one-sided limit.lim x0
12e
2x

The function f is defined as follows. (a) Find (if it exists). (b) Can the function be defined at such that it is continuous at x 0?