/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 100 Calculating limits exactly Use t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 100

Calculating limits exactly Use the definition of the derivative to evaluate the following limits. $$\lim _{x \rightarrow e} \frac{\ln x-1}{x-e}$$

Short Answer

Expert verified
Answer: The limit of the function as x approaches e is \(\frac{1}{e}\).

Step by step solution

01

Identify if the function is continuous and differentiable

In this case, the function is the natural logarithm of x. Since the natural logarithm function is continuous and differentiable for positive values of x, we can proceed to the next step.
02

Use the definition of the derivative to rewrite the limit

The definition of the derivative is: $$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$ In our case, \(f(x) = \ln x\), \(a = e\), and \(f(a) = \ln e \). Therefore, the limit can be rewritten as: $$f'(e) = \lim_{x \to e} \frac{\ln x - \ln e}{x - e}$$
03

Simplify the limit expression

Recall that the natural logarithm of e is equal to 1. So, we can rewrite the limit expression as: $$f'(e) = \lim_{x \to e} \frac{\ln x - 1}{x - e}$$
04

Using L'Hopital's Rule

The limit expression is in the indeterminate form \(\frac{0}{0}\), thus we can apply L'Hopital's Rule. L'Hopital's Rule states that, if the limit approaches an indeterminate form, we can find the limit by taking the derivative of both the numerator and denominator separately and evaluating the limit of the new function. Therefore, $$\lim_{x \rightarrow e} \frac{\ln x-1}{x-e} = \lim_{x \rightarrow e} \frac{(d/dx)(\ln x - 1)}{(d/dx)(x-e)}$$
05

Find the derivatives of the numerator and denominator

Compute the derivatives of the numerator and denominator: $$(d/dx)(\ln x - 1) = \frac{1}{x}$$ $$(d/dx)(x-e) = 1$$ Therefore, our new limit expression is: $$\lim_{x \rightarrow e} \frac{\frac{1}{x}}{1}$$
06

Evaluate the limit

Plug in \(e\) for \(x\) in the simplified limit expression: $$\lim_{x \rightarrow e} \frac{\frac{1}{x}}{1} = \frac{1}{e}$$ So, the limit of the given function as x approaches e is \(\frac{1}{e}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use the properties of logarithms to simplify the following functions before computing \(f^{\prime}(x)\). $$f(x)=\ln (3 x+1)^{4}$$

Suppose \(f\) is differentiable on an interval containing \(a\) and \(b\), and let \(P(a, f(a))\) and \(Q(b, f(b))\) be distinct points on the graph of \(f\). Let \(c\) be the \(x\) -coordinate of the point at which the lines tangent to the curve at \(P\) and \(Q\) intersect, assuming that the tangent lines are not parallel (see figure). a. If \(f(x)=x^{2},\) show that \(c=(a+b) / 2,\) the arithmetic mean of \(a\) and \(b\), for real numbers \(a\) and \(b\) b. If \(f(x)=\sqrt{x}\), show that \(c=\sqrt{a b}\), the geometric mean of \(a\) and \(b\), for \(a > 0\) and \(b > 0\) c. If \(f(x)=1 / x,\) show that \(c=2 a b /(a+b),\) the harmonic mean of \(a\) and \(b,\) for \(a > 0\) and \(b > 0\) d. Find an expression for \(c\) in terms of \(a\) and \(b\) for any (differentiable) function \(f\) whenever \(c\) exists.

Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure they actually lie on the curve. Confirm your results with a graph. $$x^{2}(y-2)-e^{y}=0$$

Horizontal tangents The graph of \(y=\cos x \cdot \ln \cos ^{2} x\) has seven horizontal tangent lines on the interval \([0,2 \pi] .\) Find the approximate \(x\) -coordinates of all points at which these tangent lines occur.

Identifying functions from an equation. The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for \(y\) to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots\) c. Use the functions found in part (b) to graph the given equation. $$y^{2}(x+2)=x^{2}(6-x) \text { (trisectrix) }$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.