2.2.1 Evaluating Limits Flashcards Preview

AP Calculus AB > 2.2.1 Evaluating Limits > Flashcards

Flashcards in 2.2.1 Evaluating Limits Deck (17)
Loading flashcards...
1
Q

Evaluating Limits

A
  • 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.
2
Q

limit

A
  • 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.
3
Q

indeterminate form

A
  • 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.
4
Q

Evaluate lim x→2 2x.

A

4

5
Q

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)

A

{−2, 0}

6
Q

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

A

f∘g

7
Q

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:

A

There is not enough information.

8
Q

Evaluate.lim t→4 13t−52/2t−8

A

13/2

9
Q

Expressions of the form 0/0 are known as:

A

Indeterminate forms

10
Q

Evaluate lim x→−2 (4x^2+1).

A

17

11
Q

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

A

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

12
Q

Evaluate.

lim x→0 (5x^3−5x^2+5−e^x)

A

4

13
Q

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.

A

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

14
Q

Evaluate.lim s→2 4s^2−4s+2

A

3

15
Q
Consider the piecewise function f(x)={|x|, x≠0
                                                              0,​   x=0  .
Is f(x) continuous at x=0?
A

Yes, the function is continuous at x = 0.

16
Q

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

A

0

17
Q

Evaluate lim t→1 t^3−t/t^2−1.

A

1

Decks in AP Calculus AB Class (190):