2.2.1 Evaluating Limits

Question 1

Evaluating Limits

Answer 1

• The limit of a function is the range value that the function approaches as you get closer to a particular domain value.
• To evaluate a limit at a value where a function is well behaved, substitute the value into the function expression.
• Limits that produce indeterminate forms may or may not exist. An indeterminate form is a signal that more work is needed to evaluate the limit.

Question 2

limit

Answer 2

• Limits allow you to study the behavior of a function near a certain x-value. If the function approaches the same value on either side of that x-value, then the limit exists.
• This limit is read as “the limit as x approaches 5 of f of x.”
• You can evaluate limits of well behaved functions by substituting the x-value into the limit expression.
• Notice that the value of the function given by y = 2x + 1 at x = 3 is the same as the limit as x approaches 3 of 2x + 1.

Question 3

indeterminate form

Answer 3

• For some limits, direct substitution will result in an indeterminate form such as 0/0. This expression cannot be evaluated since division by 0 is not defined.
• An indeterminate form is a sign that you need to do more work.
• In this case, you can factor the expression and cancel the x in the numerator with the x in the denominator. You can then substitute 0 in for each occurrence of x and determine the value of the limit. This limit is 1, which agrees with the graph of the function.
• When you cancel you have to promise that the denominator will never be 0. However, the limit is studying the function near x = 0 and not at that value. Therefore direct substitution is allowed.

Question 4

Evaluate lim x→2 2x.

Answer 4

4

Question 5

Which of the following sets represents all of the possible removable discontinuities of the function f (x) = x^ 3 + 3x^ 2 + 2x / x^ 3 + x ^2 − 2x?
(The sets represent x-values)

Answer 5

{−2, 0}

Question 6

Let f and g be continuous at a. Which of the following functions is not necessarily continuous at a ?

Answer 6

f∘g

Question 7

Suppose you are evaluating the limit lim x→5 f(x) and plugging in x=5 results in the expression 00.In this case, the value of lim x→5f(x) is:

Answer 7

There is not enough information.

Question 8

Evaluate.lim t→4 13t−52/2t−8

Answer 8

13/2

Question 9

Expressions of the form 0/0 are known as:

Answer 9

Indeterminate forms

Question 10

Evaluate lim x→−2 (4x^2+1).

Answer 10

17

Question 11

Consider the function
f(x)=x^3+x/x .
Is  f (x) continuous at x = 0?

Answer 11

No,  f (x) is not continuous at x = 0.

Question 12

Evaluate.

lim x→0 (5x^3−5x^2+5−e^x)

Answer 12

4

Question 13

Gary is simplifying the expression for a function f (x).
What is wrong with his work?

f(x)=x^2−5x+6/3x−6 = (x−2)(x−3)/3(x−2) = x−3/3

Therefore, Gary concludes,
f(x)=x−3/3.

Answer 13

Gary has to note that x cannot equal 2 using his simplified expression.

Question 14

Evaluate.lim s→2 4s^2−4s+2

Answer 14

3

Question 15

Consider the piecewise function f(x)={|x|, x≠0
                                                              0,​   x=0  .
Is f(x) continuous at x=0?

Answer 15

Yes, the function is continuous at x = 0.

Question 16

Evaluate the following limit. lim |x| x→0

Answer 16

0

Question 17

Evaluate lim t→1 t^3−t/t^2−1.

Answer 17

1