Calculus Homework Assignment 2

Class:

Student Number:

Name:

1. Evaluate the limit, if it exists.

a. limx→−2

2x2 + x − 6x3 + 8

.

b. limt→0

(1

t 3√

1 + t− 1

t

). [like §2.3 #19, 29]

3.

a. If limx→1

f(x) + 3x2 + x − 2

= 10, find limx→1

f(x).

b. Find the value of a such that the limit

limx→−2

3x3 + ax2 + 4x2 + (2 − a)x − 2a

exits, and evaluate the limit. [like §2.3 #55, 61]

2. Let f(x) = x − 2⌊x⌋ where the symbol ⌊ ⌋denotes the greatest integer function and let nbe an integer.a. Evaluate lim

x→n+f(x) and lim

x→n−f(x).

b. For what values of a does limx→a

f(x) exists?

[like §2.3 #49]

4. Evaluate the limit limx→2

√x and find the value

of δ that correspond to ϵ = 0.1.[like §2.4 #14]

(Over Please)

Calculus Homework Assignment 2

5. Prove that limx→a

3√

x = 3√

a. [like §2.4 #37] 7. Show that the function

f(x) =

{x sin

1x

if x ̸= 00 if x = 0

is continuous on (−∞,∞).[like §2.5 #63]

6. Find the values of a and b that make f con-tinuous everywhere.

f(x) =

√

x + 2 − 1x + 1

if x < −1

ax3 + bx2 + 3 if −1 ≤ x < 2bx2 + 7x + 2a + 1 if x ≥ 2

[like §2.5 #42]

8.a. Show that the absolute value functionF (x) = |x| is continuous everywhere.b. Prove that if f is a continuous function ev-erywhere, then so is |f |.c. Is the converse of the statement in b. alsotrue? In other words, if |f | is continuous, doesit follow that f is continuous? If so, prove it. Ifnot, find a counterexample. [§2.5 #64]