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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 33a
Chapter 2, Problem 33a

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


a. The value of limx→3x2−9x−3{\(\displaystyle\]\lim\)_{x\(\to\)3}}\(\frac{x^2-9}{x-3}\) does not exist.

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Identify the limit expression: \( \lim_{x \to 3} \frac{x^2 - 9}{x - 3} \).
Notice that direct substitution of \( x = 3 \) results in a \( \frac{0}{0} \) indeterminate form.
Factor the numerator: \( x^2 - 9 = (x - 3)(x + 3) \).
Rewrite the limit expression: \( \lim_{x \to 3} \frac{(x - 3)(x + 3)}{x - 3} \).
Cancel the common factor \( x - 3 \) in the numerator and denominator, then evaluate the limit of the simplified expression.

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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 points of interest, including points where they may not be defined. For example, the limit of a function can exist even if the function itself does not take a value at that point.
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Indeterminate Forms

Indeterminate forms arise in calculus when evaluating limits that do not lead to a clear value, such as 0/0 or ∞/∞. These forms require further analysis, often using algebraic manipulation or L'Hôpital's Rule, to resolve the limit. Recognizing these forms is crucial for determining the existence of limits in complex expressions.
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Factoring and Simplifying Expressions

Factoring and simplifying expressions is a technique used to rewrite complex algebraic expressions in a more manageable form. In the context of limits, this often involves canceling common factors to eliminate indeterminate forms. For instance, the expression (x^2 - 9)/(x - 3) can be factored to (x - 3)(x + 3)/(x - 3), allowing for simplification and easier limit evaluation.
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Related Practice
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

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


lim x→2 (5x−6)^3/2

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Textbook Question

Determine the following limits. 


lim x→∞ 6x2/(4x^2+√(16x4 + x2))

Textbook Question

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


lim x→1 x^2 − 1 / x − 1

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Textbook Question

Determine the following limits. 


lim x→−∞ (x+ √x^2−5x)

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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)