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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 2.45
Chapter 2, Problem 2.45

Determine the following limits.
lim w→∞ (ln w2) / (ln w3 + 1)

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Identify the limit expression: \( \lim_{w \to \infty} \frac{\ln w^2}{\ln w^3 + 1} \).
Simplify the expression: \( \ln w^2 = 2 \ln w \) and \( \ln w^3 = 3 \ln w \).
Rewrite the limit using the simplified expressions: \( \lim_{w \to \infty} \frac{2 \ln w}{3 \ln w + 1} \).
Factor out \( \ln w \) from the denominator: \( \lim_{w \to \infty} \frac{2 \ln w}{\ln w (3 + \frac{1}{\ln w})} \).
Cancel \( \ln w \) and evaluate the limit: \( \lim_{w \to \infty} \frac{2}{3 + \frac{1}{\ln w}} \).

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Key Concepts

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

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. In this case, we are interested in the limit of a function as w approaches infinity. Understanding limits helps in analyzing the asymptotic behavior of functions and is crucial for evaluating expressions that may not be directly computable.
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One-Sided Limits

Natural Logarithm

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is a key function in calculus, particularly in growth and decay problems. In the given limit, the natural logarithm of w raised to a power is involved, which simplifies to a multiplication of the exponent and ln(w), illustrating the properties of logarithms.
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Derivative of the Natural Logarithmic Function

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful in the context of the given limit, as it can simplify the evaluation of the logarithmic expressions.
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Related Practice
Textbook Question

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.

f(x)=1/ √x sec x

Textbook Question

Evaluate limx→∞f(x){\(\displaystyle\[\lim\)_{x\(\to\]\infty\)}{f(x)}} andlimx→−∞f(x){\(\displaystyle\)\(\lim\)_{x\(\to\)-\(\infty\)}{f(x)}}.


f(x)=4x3+11−x3f\(\left\)(x\(\right\))=\(\frac{4x^3+1}{1-x^3}\)

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Textbook Question

Describe the end behavior of g(x) = e-2x.

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Textbook Question

Use a graph of f to estimate limx→af(x){\(\displaystyle\)\(\lim\)_{x\(\to\) a}f\(\left\)(x\(\right\))} or to show that the limit does not exist. Evaluate f(x) near x=ax=a to support your conjecture.

f(x)=x−2ln∣x−2∣f\(\left\)(x\(\right\))=\(\frac{x-2}{\ln\left|x-2\right|}\); a=2a=2

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Textbook Question

Evaluate each limit and justify your answer. 

lim x→0 (x^8−3x^6−1)^40

Textbook Question

Suppose limx→a f(x)=L{\(\displaystyle\)\(\lim\)_{x\(\to\) a}}\(\text{ }\)f\(\left\)(x\(\right\))=L and limx→a g(x)=M{\(\displaystyle\)\(\lim\)_{x\(\to\) a}}\(\text{ }\)g\(\left\)(x\(\right\))=M. Prove that limx→a (f(x)−g(x))=L−M{\(\displaystyle\[\lim\)_{x\(\to\) a}}\(\text{ }\]\left\)(f\(\left\)(x\(\right\))-g\(\left\)(x\(\right\))\(\right\))=L-M.

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