1. Limits and Continuity

At what points are the functions in Exercises 13–30 continuous?

y = (2x – 1)¹/³

Determine the interval(s) for which the function is continuous.

$f\left(x\right)=x^2+4x-12$

Limits and Continuity

On what intervals are the following functions continuous?

c. h(x) = x⁻²/³

Determine the interval(s) for which the function is continuous.

Complete the following steps for each function.

c. State the interval(s) of continuity.

f(x)={x^3+4x+1 if x≤0

2x^3 if x>0; a=0

Use continuity to evaluate the limit: $\underset{x\to 1}{lim}\frac{ln(7-x^2)}{1+x}$

Use the continuity of the absolute value function (Exercise 78) to determine the interval(s) on which the following functions are continuous.

$g\left(x\right)=\mid \frac{x+4}{x^2-4}\mid$

Let g(x)= {1 if x≥0

−1 if x<0.

a. Write a formula for |g(x)|.

For what value of the constant $c$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}2x+1& \text{, }x<3\\ c& \text{, }x=3\\ x^2-2& \text{, }x>3\end{array}\right\}$$ continuous on $ℝ$?

Suppose you park your car at a trailhead in a national park and begin a 2-hr hike to a lake at 7 A.M. on a Friday morning. On Sunday morning, you leave the lake at 7 A.M. and start the 2-hr hike back to your car. Assume the lake is 3 mi from your car. Let f(t) be your distance from the car t hours after 7 a.m. on Friday morning, and let g(t) be your distance from the car t hours after 7 a.m. on Sunday morning.

b. Let h(t)=f(t)−g(t). Find h(0) and h(2).

Determine the interval(s) on which the following functions are continuous; then analyze the given limits. 

f(x)=csc x;lim x→π/4f (x);lim x→2π^− f(x)

Use continuity to evaluate the limit: $\underset{x\to 2}{lim}3x^2-5x+4$.

Complete the following sentences in terms of a limit.

a. A function is continuous from the left at a if _____.

Suppose the rental cost for a snowboard is \$25 for the first day (or any part of the first day) plus \$15 for each additional day (or any part of a day).

e. For what values of t is f continuous? Explain.

Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. 

f(x)=2x^2+3x+1 / x^2+5x; a=−5