1. Limits and Continuity

Define h(2) in a way that extends h(t) = (t² + 3t − 10)/(t − 2) to be continuous at t = 2.

Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right? 

f(z)=(z−1)^3/4

For what values of a and b is

g(x) = { ax + 2b, x ≤ 0

x² + 3a – b, 0 < x ≤ 2

3x – 5, x > 2

continuous at every x?

Determine the value(s) of $x$ (if any) for which the function is discontinuous.

Find the intervals on which the following functions are continuous. Specify right- or left-continuity at the finite endpoints.

$h\left(x\right)=\frac{2x}{x^3-25x}$

The sign-preserving property of continuous functions Let f be defined on an interval (a, b) and suppose that f(c) ≠ 0 at some c where f is continuous. Show that there is an interval (c − δ, c + δ) about c where f has the same sign as f(c).

Never-zero continuous functions Is it true that a continuous function that is never zero on an interval never changes sign on that interval? Give reasons for your answer.

Determine the interval(s) on which the following functions are continuous. 

p(x)=3x^2−6x+7 / x^2+x+1

Which of the following functions are continuous for all values in their domain? Justify your answers.

a. a(t)=altitude of a skydiver t seconds after jumping from a plane

Let $g\(\left\)(x\(\right\))=\(\begin{cases}\)x^2+x & \(\text{if }\)x<1\\ a & \(\text{if }\)x=1\\ 3x+5 & \(\text{if }\)x>1\(\end{cases}\)$

a. Determine the value of a for which $g$ is continuous from the left at $1$.

Use the graph of $f\(\left\)(x\(\right\))$ to determine if the function is continuous or discontinuous at $x=c$.

$c=4$

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

y = 1/(|x| + 1) − x²/2

Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right? 

f(t)=(t^2−1)^3/2

Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?

lim x → 0 sin ((π + tan x)/(tan x – 2 sec x))

[Technology Exercise] In Exercises 33–36, graph the function to see whether it appears to have a continuous extension to the given point a. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at a. If the function does not appear to have a continuous extension, can it be extended to be continuous from the right or left? If so, what do you think the extended function’s value should be?

g(θ) = 5 cos θ / (4θ ― 2π) , a = π/2