2.2.2 Limits and Indeterminate Forms

Question 1

Limits and Indeterminate Forms

Answer 1

• 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.
• If direct substitution produces zero divided by a non-zero number, then the limit equals zero. If it produces a non-zero number divided by zero, the limit is undefined.

Question 2

note

Answer 2

• Always try direct substitution as your first step when evaluating a limit.
• In this example direct substitution leads to the value of limit 2.
• When you first use direct substitution with this limit, it will produce the indeterminate form0/0. Try factoring the numerator and denominator. When you do, you will find that they have a common factor of x– 5. Cancel the common factors, keeping in mind that this is a limit.
• The resulting expression can be evaluated using direct substitution.
• Before you begin factoring, make sure you really have an indeterminate form. Zero divided by any number other than zero is not indeterminate. It equals zero.
• In addition, any non-zero number divided by zero is not indeterminate. It is undefined. If a limit produces this type of expression, then the limit does not exist.
• Here is a summary of quotients involving zero. Only the first one is an indeterminate form.

Question 3

Evaluate the limit lim x→−2 1/(2+x)^3.

Answer 3

The limit does not exist

Question 4

Evaluate lim x→2 x+1/x^2−x−2.

Answer 4

The limit does not exist

Question 5

Given g(x)= b−|x−a|,
h(x )= b+|x−a|,
and g(x) ≤ f(x) ≤h(x),
find lim x→a f(x).

Answer 5

b

Question 6

Given g(x)= 2−x^2,
h(x)=2+x^2,
and g(x)≤f(x)≤h(x),
find lim x→0 f(x).

Answer 6

2

Question 7

Suppose that you are evaluating a limit, and after some simplification you reach the expression a0, where a≠0. What is the value of the limit?

Answer 7

The limit does not exist

Question 8

Evaluate the limit lim x→−1 2/(x+1)^3.

Answer 8

The limit does not exist

Question 9

Evaluate lim x→−5 x+5/x^2+7x+10.

Answer 9

−1/3

Question 10

Evaluate lim x→3/2 8x^3−27/2x−3.

Answer 10

27

Question 11

Evaluate lim x→−1/2 8x^2−2x−3/6x^2+x−1.

Answer 11

2

Question 12

Evaluate limx→−3 x^2+5x+6/x+3.

Answer 12

-1