Question

Find the limit, if it exists. $$ \lim _{x \rightarrow 2} \frac{\ln (x-1)}{x-2} $$

   Find the limit, if it exists.
$$
\lim _{x \rightarrow 2} \frac{\ln (x-1)}{x-2}
$$

Calculus

Earl W. Swokowski 5th Edition

Chapter 10, Problem 24

Instant Answer

Solved by Expert Gregory Higby

03/22/2022

Step 1

We can try to do this by direct substitution, but we quickly see that this results in an indeterminate form of $\frac{0}{0}$, because $\ln(2-1) = 0$ and $2-2 = 0$. Show more…

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Find the limit, if it exists. $$ \lim _{x \rightarrow 2} \frac{\ln (x-1)}{x-2} $$

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Key Concepts

Derivative of the Natural Logarithm

Understanding the derivative of the natural logarithm is key in calculus, especially when applying L'Hôpital's Rule. The derivative of ln(u) with respect to u is 1/u, which simplifies the process of evaluating limits involving logarithmic expressions when direct substitution leads to an indeterminate form.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool used to evaluate limits involving indeterminate forms, particularly the 0/0 and ?/? types. It states that if the limits of the numerator and denominator both approach zero or both approach infinity, the limit of the original quotient can be found by taking the derivatives of the numerator and denominator separately and then taking the limit of the new quotient.

Limit

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain point. It investigates the value that a function tends toward as the variable nears a specific value, which is essential for understanding continuity, derivatives, and the overall behavior of functions.

Indeterminate Forms

An indeterminate form arises in limit calculations when substituting a value into a function yields an undefined expression, such as 0/0 or ?/?. This signals that the limit cannot be directly computed and requires methods like algebraic manipulation or advanced techniques to resolve the ambiguity.

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