1. Limits and Continuity

Limits and Infinity

Find the limits in Exercises 37–46.

x²/³ + x⁻¹

lim --------------------

x→∞ x²/³ + cos²x

Horizontal and Vertical Asymptotes

Use limits to determine the equations for all horizontal asymptotes.

_____

√x² + 4

c. g(x) = -----------

x

Horizontal and Vertical Asymptotes

Use limits to determine the equations for all horizontal asymptotes.

_________

/ x² + 9

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

√ 9x² + 1

Horizontal and Vertical Asymptotes

Assume that constants a and b are positive. Find equations for all horizontal and vertical asymptotes for the graph of y = (√ax² + 4) / (x―b) .

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) |

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)

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 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

Finding Limits

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

lim x →π sin (x/2 + sin x)

Finding Limits

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

lim x →π cos² (x― tan x)