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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 2.4.36
Chapter 2, Problem 2.4.36

Determine the following limits.


limz→4z−5(z2−10z+24)2{\(\displaystyle\]\lim\)_{z\(\to\)4}}\(\frac{z-5}{\left(z^2-10z+24\right)^2}\)

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Step 1: Identify the limit expression: \( \lim_{{z \to 4}} \frac{{z-5}}{{(z^2 - 10z + 24)^2}} \).
Step 2: Factor the quadratic expression in the denominator: \( z^2 - 10z + 24 \). This can be factored as \((z-4)(z-6)\).
Step 3: Substitute the factored form into the limit expression: \( \lim_{{z \to 4}} \frac{{z-5}}{{((z-4)(z-6))^2}} \).
Step 4: Notice that direct substitution of \( z = 4 \) results in an indeterminate form \( \frac{0}{0} \). Apply L'Hôpital's Rule or simplify the expression by canceling common factors.
Step 5: Simplify the expression by canceling the common factor \( z-4 \) from the numerator and the denominator, then re-evaluate 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. In this case, we are interested in the limit of a fraction as the variable z approaches 4.
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Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to obtain the original polynomial. This is crucial for simplifying expressions, especially when evaluating limits, as it can help eliminate indeterminate forms like 0/0. In the given limit, factoring the denominator will be necessary to simplify the expression.
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Indeterminate Forms

Indeterminate forms occur in calculus when evaluating limits leads to expressions that do not have a clear value, such as 0/0 or ∞/∞. These forms require further analysis, often through algebraic manipulation or L'Hôpital's rule, to resolve. In this limit problem, substituting z = 4 directly results in an indeterminate form, necessitating further steps to find the limit.
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Related Practice
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.

lim x→2 (x^2+3x)=10

Textbook Question

Determine limx→∞f(x)\(\lim\)_{x\(\rightarrow\]\infty\)}f\(\left\)(x\(\right\)) and limx→−∞f(x)\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\)) for the following functions. Then give the horizontal asymptotes of ff (if any).


f(x)=4x(3x−9x2+1)f\(\left\)(x\(\right\))=4x\(\left\)(3x-\(\sqrt{9x^2+1}\]\right\))

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Textbook Question

Determine the following limits.

limt→∞cos(t)e3t{\(\displaystyle\[\lim\)_{t\(\to\]\infty\)}\(\frac{\cos\left(t\right)}{e^{3t}\)}}

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Textbook Question

Evaluate each limit and justify your answer. 

lim t→4 t−4 /√t−2

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Textbook Question

Determine the following limits. 

lim x→−∞ (3x7 + x2) 

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers and assume lim x→a f(x) =L


d. If |x−a|<δ, then a−δ<x<a+δ.