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AP Calculus AB: 2.2.1 Evaluating Limits

Mathematics17 CardsCreated 3 months ago

To evaluate a limit, substitute the x-value into the function if it's continuous there. If substitution gives an indeterminate form like 0/0, use algebraic methods (like factoring or simplifying) to resolve it. Limits describe a function’s behavior near a point—not necessarily at the point.

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Evaluating Limits

  • 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.

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Key Terms

Term
Definition

Evaluating Limits

  • The limit of a function is the range value that the function approaches as you get closer to a particular domain value.

  • To e...

limit

  • 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 th...

indeterminate form

  • For some limits, direct substitution will result in an indeterminate form such as 0/0. This expression cannot be evaluated since division b...

Evaluate lim x→2 2x.

4

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)

{−2, 0}

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

f∘g

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AP Calculus AB: 2.2.1 Evaluating Limits
TermDefinition

Evaluating Limits

  • 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.

limit

  • 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.

indeterminate form

  • 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.

Evaluate lim x→2 2x.

4

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)

{−2, 0}

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

f∘g

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:

There is not enough information.

Evaluate.lim t→4 13t−52/2t−8

13/2

Expressions of the form 0/0 are known as:

Indeterminate forms

Evaluate lim x→−2 (4x^2+1).

17

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

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

Evaluate.

lim x→0 (5x^3−5x^2+5−e^x)

4

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.

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

Evaluate.lim s→2 4s^2−4s+2

3

Consider the piecewise function f(x)={|x|, x≠0

0,​   x=0  .

Is f(x) continuous at x=0?

Yes, the function is continuous at x = 0.

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

0

Evaluate lim t→1 t^3−t/t^2−1.

1