Limits and Continuity

On what intervals are the following functions continuous?

c. h(x) = cos x / x―π

Suppose limx→4 f(x) = 0 and lim x→4 g(x) = −3. Find

c. limx→4 (g(x))²

Horizontal and Vertical Asymptotes

Use limits to determine the equations for all horizontal asymptotes.

_________

/ x² + 9

d. y = / -------------

√ 9x² + 1

Finding Limits Graphically

Which of the following statements about the function y = f(x) graphed here are true, and which are false?

d. limx→1− f(x) = 2

Limits and Continuity

On what intervals are the following functions continuous?

d. k(x) = sin x / x

Finding Limits Graphically

Graph the functions in Exercises 9 and 10. Then answer these questions.

f(x) = {x,−1 ≤ x < 0, or 0 < x ≤ 1

1, x = 0

0, x < −1 or x > 1

d. At what points does the right-hand limit exist but not the left-hand limit?

Limits and Continuity

On what intervals are the following functions continuous?

d. k(x) = x⁻¹/⁶

Finding Limits Graphically

Which of the following statements about the function y = f(x) graphed here are true, and which are false?

e. limx→1+ f(x) = 1

Finding Limits Graphically

Which of the following statements about the function y = f(x) graphed here are true, and which are false?

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g. limx→0+ f(x) = limx→0− f(x)

Finding Limits

For the function f whose graph is given, determine the following limits. Write ∞ or −∞ where appropriate.

h. lim x → ∞ f(x)

Which of the following statements about the function y=f(x) graphed here are true, and which are false?

h. f(0)=0

Finding Limits Graphically

Which of the following statements about the function y = f(x) graphed here are true, and which are false?

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k. limx→3+ f(x) does not exist.

Limits and Continuity

Suppose that ƒ(t) and ƒ(t) are defined for all t and that lim t → t₀ ƒ(t) = ―7 and lim (t → t₀) g (t) = 0 . Find the limit as t → t₀ of the following functions.

e. cos (g(t))

Limits and Continuity

Suppose that ƒ(t) and ƒ(t) are defined for all t and that lim t → t₀ ƒ(t) = ―7 and lim (t → t₀) g (t) = 0 . Find the limit as t → t₀ of the following functions.

f. | ƒ(t) |

Which of the following statements about the function y=f(x) graphed here are true, and which are false?

g. limx→1 f(x) does not exist.

Limits and Continuity

Suppose that ƒ(t) and ƒ(t) are defined for all t and that lim t → t₀ ƒ(t) = ―7 and lim (t → t₀) g (t) = 0 . Find the limit as t → t₀ of the following functions.

h. 1 / ƒ(t)

Which of the following statements about the function y=f(x) graphed here are true, and which are false?

i. f(0)=1

Which of the following statements about the function y=f(x) graphed here are true, and which are false?

a. limx→2 f(x) does not exist.

Which of the following statements about the function y=f(x) graphed here are true, and which are false?

b. limx→2 f(x)=2

Which of the following statements about the function y=f(x) graphed here are true, and which are false?

c. limx→1 f(x) does not exist.

Which of the following statements about the function y=f(x) graphed here are true, and which are false?

d. limx→c f(x) exists at every point c in (-1,1).

Limits and Continuity

Suppose the functions ƒ(x) and g(x) are defined for all x and that lim (x → 0) ƒ(x) = 1/2 and lim (x → 0) g(x) = √2. Find the limits as x → 0 of the following functions.

e. x + ƒ(x)

Limits and Continuity

Suppose the functions ƒ(x) and g(x) are defined for all x and that lim (x → 0) ƒ(x) = 1/2 and lim (x → 0) g(x) = √2. Find the limits as x → 0 of the following functions.

f. [ƒ(x) • cos x ] / x―1

Limits and Continuity

In Exercises 5 and 6, find the value that lim (x→0) g(x) must have if the given limit statements hold.

lim ((4―g(x)) / x ) = 1

x→0

Limits and Continuity

In Exercises 5 and 6, find the value that lim (x→0) g(x) must have if the given limit statements hold.

lim (x lim g(x)) = 2

x→-4 x→0

Finding Deltas Graphically

In Exercises 7–14, use the graphs to find a δ>0 such that ​|f(x)−L| <ε whenever 0< |x−c| <δ.

Finding Limits

In Exercises 9–24, find the limit or explain why it does not exist.

lim x→a (x² ― a²)/(x⁴ ― a⁴)

Finding Limits

In Exercises 9–24, find the limit or explain why it does not exist.

lim h →0 ((x + h)² ― x²)/h

The accompanying figure shows the plot of distance fallen versus time for an object that fell from the lunar landing module a distance 80 m to the surface of the moon.

a. Estimate the slopes of the secant lines PQ₁, PQ₂, PQ₃, and PQ₄, arranging them in a table like the one in Figure 2.6.

b. About how fast was the object going when it hit the surface?

Finding Limits

In Exercises 9–24, find the limit or explain why it does not exist.

lim x →π sin (x/2 + sin x)