Section 1.8 Homework

 

1. From the graph of f, state each x-value at which f is discontinuous. For each x-value, determine whether f is continuous from the right, or from the left, or neither. (Enter your answers from smallest to largest.)

x= -4

x= -2

x=2

x=4

 

2. Sketch the graph of a function f that is continuous except for the stated discontinuity.
Discontinuous at 4, but continuous from the right there

 

 

3.Sketch the graph of a function f that is continuous except for the stated discontinuity.
Removable discontinuity at 4, jump discontinuity at 6

 

 

4. Explain why the function is discontinuous at the given number a. (Select all that apply.)

$f\left(x\right)=\frac{1}{x+3} a=–3$

 

 

Sketch the graph of the function.

 

 

5. Explain why the function is discontinuous at the given number a. (Select all that apply.)

 

Sketch the graph of the function.

 

6.  How would you “remove the discontinuity” of f ? In other words, how would you define f(5)  in order to make f continuous at 5?

$f\left(x\right) = \frac{x^2–2x–15}{x–5}$

f(5) =

 

7.   Use continuity to evaluate the limit.

$\underset{x\to 4}{\lim }x\sqrt{32–x^2}$

 

 

8.  Use continuity to evaluate the limit.

$\underset{x\to \mathrm{\pi }}{\lim }8 \sin (x + \sin (x\left)\right)$

 

 

9. Find each x-value at which f is discontinuous and for each x-value, determine whether f is continuous from the right, or from the left, or neither.

 

 

x=0

 

x=1

 

Sketch the graph of f.

 

 

10.  For what value of the constant c is the function f continuous on (−∞, ∞)?

 

c=

 

 

11.  Find the values of a and b that make f continuous everywhere.

 

a=

b=

 

 

12.  Which of the following functions f has a removable discontinuity at a? If the discontinuity is removable, find a function g that agrees with f for x ≠ a and is continuous at a. (If an answer does not exist, enter DNE.)

 

a.

$f\left(x\right)=\frac{x^4–1}{x–1}, a=1$

 

g(x) = ___________________

 

b.

$f\left(x\right) = \frac{x^3–x^2–20x}{x–5}, a=5$

 

g(x) = ___________________

 

 

c.

$f\left(x\right)=\left[\right[\sin \left(x\right)\left]\right], a=\mathrm{\pi } (\mathrm{Recall} \mathrm{that} [\left[\mathrm{h}\right(\mathrm{x}\left)\right]]$

means the largest integer that is less than or equal to h(x).)

 

g(x) =  ____________________________