2.2.4 An Overview of Limits

Question 1

An Overview of Limits

Answer 1

• The limit is the range value that a function approaches as you get closer to a particular domain value.
• An indeterminate form is a mathematically meaningless expression.

Question 2

note

Answer 2

• This limit involves an unusual variable.
• Remember to use direct substitution as a first step in evaluating limits. In this case, direct substitution produces the familiar indeterminate formof 0/0.
• Proceed by factoring the numerator, which is a difference of two squares.
• Use cancellation to simplify the limit expression and then apply direct substitution to arrive at the result.
• The existence of limits can be demonstrated graphically. On the far left, the graph shows that near x= 7 the function is approaching the same value from both the left and the right. The limit exists and equals that value, even though the function takes on a different value at x= 7.
• On the near left, the graph approaches different values on either side of x= 5. Since the two one-sided limits have different values, the limit of the function does not exist.
• Here is an example of a function that is approaching very large values from the one side and very small values from the other. The limit for such a function does not exist.

Question 3

LetG(x)= x^2−4/x+2, x≠−2
k, x=−2
Find the value of k so that lim x→−2 G(x)=G(−2).

Answer 3

-4

Question 4

Classify all of the discontinuities of the function h(x)=f(g(x)) given f(x)=1/x−3 and g(x)=x^2+2.

Answer 4

x = −1 and x = 1; infinite discontinuities

Question 5

Given that lim x→0(sinx)^2/x=0, find the limit.

lim x→0 1−cosx/x

Answer 5

0

Question 6

Evaluate the limit limCOW→3

[4(COW)−12 / (COW)^2+(COW)−12].

Answer 6

4/7

Question 7

Does f (x) have a limit at x = −3?

Answer 7

No, the limit doesn’t exist.

Question 8

Evaluate the limit

limΔx→0 4(Δx+2)^2+5Δx−3/6Δx+1

Answer 8

13

Question 9

If f(x)=4x2−4xx+1,
evaluate the limit lim x→−1 f(x).

Answer 9

The limit does not exist.

Question 10

Given the limit lim x→2(2x+2)=6, what is the largest value of δ such that ε

Answer 10

.005

Question 11

Given the limit lim x→1 (4x+3)=7,what is the largest value of δ such that ε≤.01?

Answer 11

.0025