Exponential limit Prove that lim(1+) = ", for a + 0.%3D

Given: prove that limx→∞1+axx=ea, a≠0 for evaluating given limit, we solve left hand side for that…

Answered: Exponential limit Prove that lim (1+) =…

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

Chapter2: Graphical And Tabular Analysis

Section2.FR1: A Further Look: Limits

Problem 3E

See similar textbooks

Related questions

Question

Exponential limit Prove that lim
(1+) = ", for a + 0.
%3D

Expert Solution

Step 1

Given: prove that $\lim_{x \to \infty} {(1 + \frac{a}{x})}^{x} = e^{a}, a \neq 0$

for evaluating given limit, we solve left hand side for that we assuming given limit is equal to L

so,

$L = \lim_{x \to \infty} {(1 + \frac{a}{x})}^{x} . . (1)$

now, for evaluating given limit, we take log both side

Step 2

so,

$\begin{array}{rcl} \ln L & = & \lim_{x \to \infty} \ln {(1 + \frac{a}{x})}^{x} (∵ \ln m^{n} = n \ln m) \\ = & \lim_{x \to \infty} [x \ln (1 + \frac{a}{x})] \\ = & \lim_{x \to \infty} \frac{\ln (1 + \frac{a}{x})}{(\frac{1}{x})} . . (2) \end{array}$

now, using L hospital's rule in equation $(2)$

$\begin{array}{rcl} \ln L & = & \lim_{x \to \infty} \frac{\ln (1 + \frac{a}{x})}{(\frac{1}{x})} \\ = & \lim_{x \to \infty} \frac{(\frac{1}{1 + \frac{a}{x}}) (0 - \frac{a}{x^{2}})}{(- \frac{1}{x^{2}})} \\ = & \lim_{x \to \infty} \frac{a}{(1 + \frac{a}{x})} \\ = & \frac{a}{1 + 0} \\ = & a \\ L & = & e^{a} = R . H . S . \end{array}$

Step by step

Solved in 3 steps

Exponential limit Prove that lim (1+) = ", for a + 0. %3D