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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 2.6.33
Chapter 2, Problem 2.6.33

Evaluate each limit and justify your answer. 
lim x→4 √x^3−2x^2−8x / x−4

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1
Identify the type of limit: This is a limit of the form \( \frac{0}{0} \) as both the numerator and denominator approach zero when \( x \to 4 \).
Apply L'Hôpital's Rule: Since the limit is of the indeterminate form \( \frac{0}{0} \), we can apply L'Hôpital's Rule, which involves differentiating the numerator and the denominator separately.
Differentiate the numerator: Find the derivative of \( \sqrt{x^3 - 2x^2 - 8x} \) with respect to \( x \). Use the chain rule and power rule for differentiation.
Differentiate the denominator: The derivative of \( x - 4 \) with respect to \( x \) is simply 1.
Evaluate the new limit: Substitute \( x = 4 \) into the new expression obtained after applying L'Hôpital's Rule and simplify 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 points of interest, including points where they may not be defined. Evaluating limits often involves techniques such as substitution, factoring, or applying L'Hôpital's rule when dealing with indeterminate forms.
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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. In such cases, further analysis is required, often involving algebraic manipulation or L'Hôpital's rule, which allows for differentiation of the numerator and denominator to resolve the limit.
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Factoring and Simplification

Factoring and simplification are techniques used to rewrite expressions in a more manageable form, especially when evaluating limits. By factoring out common terms or simplifying complex expressions, one can often eliminate indeterminate forms and make it easier to compute the limit. This process is crucial in finding the limit of rational functions or polynomials.
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Related Practice
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Determine the following limits.


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Use the precise definition of infinite limits to prove the following limits.


limx→−11(x+1)4=∞{\(\displaystyle\]\lim\)_{x\(\to\)-1}}\(\frac{1}{\left(x+1\right)^4}\)=\(\infty\)

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Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

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Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→5 1/x^2=1/25

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Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = x^2−25 / x−5; a=5

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Textbook Question

Determine the following limits. 

lim x→∞ (3x12 − 9x7)

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