1. Limits and Continuity

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)

Evaluate each limit. 

lim x→0 e^4x−1 / e^x−1

Determine the following limits.

a. $\underset{x\to 2^{+}}{\lim }\frac{1}{\sqrt{x(x-2)}}$

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim h→0 (5 + h)^2 − 25 / h

Limits of quotients

Find the limits in Exercises 23–42.

limx→1 (x −1) / (√(x + 3) − 2)

Find the limit.

$\(\lim\)_{x\(\rarr\)2}\(\frac{x^2-7x+12}{x-3}\)$

Assume postage for sending a first-class letter in the United States is \$0.47 for the first ounce (up to and including 1 oz) plus \$0.21 for each additional ounce (up to and including each additional ounce).

b. Evaluate lim w→2.3 f(w).

Limits and Infinity

Find the limits in Exercises 37–46.

sin x

lim ------------- ( If you have a grapher, try graphing

x→∞ |x| the function for ―5 ≤ x ≤ 5 ) .

Find the arc length of the graph of the function $y=\frac{2}{3}{x}^{\frac{3}{2}}+\frac{1}{3}$ over the interval $0\le x\le 9$.

Finding One-Sided Limits Algebraically

Find the limits in Exercises 11–20.

a. limx→1+ (√2x (x − 1)) / |x − 1|

Find the horizontal asymptotes of each function using limits at infinity.

f(x) = (2e^x + 3) / (e^x + 1)

Determine the following limits.

lim h→0 (h + 6)^2 + (h + 6) − 42 / h

Evaluate each limit. 

lim x→0+ 1−cos^2x / sin x

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

44. lim (x → 0⁺) (csc x - cot x + cos x)

Evaluate $\underset{x\to \infty }{\lim }f\left(x\right)$ and$\underset{x\to -\infty }{\lim }f\left(x\right)$.

$f\left(x\right)=\frac{4x^3+1}{1-x^3}$