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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 35
Chapter 2, Problem 35

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.


lim x→4 x^2 − 16 / 4 − x

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The expression \( \frac{x^2 - 16}{4 - x} \) is in an indeterminate form \( \frac{0}{0} \) as \( x \to 4 \).
Notice that \( x^2 - 16 \) is a difference of squares, which can be factored as \( (x - 4)(x + 4) \).
Substitute the factored form into the expression: \( \frac{(x - 4)(x + 4)}{4 - x} \).
Notice that \( 4 - x = -(x - 4) \), so the expression becomes \( \frac{(x - 4)(x + 4)}{-(x - 4)} \). Cancel \( x - 4 \) from the numerator and denominator, leaving \( -(x + 4) \).
Now that the expression is simplified to \( -(x + 4) \), substitute \( x = 4 \) to find the limit.>

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near specific points, which is crucial for evaluating expressions that may be undefined at those points. In this case, we are interested in the limit as x approaches 4.
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Factoring

Factoring is the process of breaking down an expression into simpler components, which can help simplify complex limits. In the given limit, the expression x^2 - 16 can be factored into (x - 4)(x + 4), allowing for cancellation of terms that may lead to an indeterminate form when directly substituting x = 4.
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Indeterminate Forms

Indeterminate forms occur when direct substitution in a limit leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. Recognizing these forms is essential, as they often require further manipulation, such as factoring or applying L'Hôpital's Rule, to evaluate the limit correctly.
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Related Practice
Textbook Question

Determine the following limits. 


lim x→−∞ (e^x cos x +3)

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


e. limx→π2cotx=0{\(\displaystyle\[\lim\)_{x\(\to\]\frac{\pi}{2}\)}}\(\cot\) x=0 . (Hint: Graph y=cot x)

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Textbook Question

Assume postage for sending a first-class letter in the United States is \$0.47 for the first ounce (up to and including 1 oz) plus \$0.21 for each additional ounce (up to and including each additional ounce).


b. Evaluate lim w→2.3 f(w).

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Textbook Question

Determine the following limits. 


lim x→−∞ (x+ √x^2−5x)

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Textbook Question

Postage rates Assume postage for sending a first-class letter in the United States is \$0.47 for the first ounce (up to and including 1 oz) plus \$0.21 for each additional ounce (up to and including each additional ounce).


a. Graph the function p=f(w) that gives the postage p for sending a letter that weighs w ounces, for 0<w≤3.5.

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


d. limx→0x{\(\displaystyle\]\lim\)_{x\(\to\)0}}\(\sqrt{x}\) . (Hint: Graph y=√x)