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Chapter 8: Problem 115

Evaluate \(\lim _{x \rightarrow \infty}\left[\frac{1}{x} \cdot \frac{a^{x}-1}{a-1}\right]^{1 / x}\) where \(a>0, a \neq 1\)

Short Answer

Expert verified
The evaluation of \(\lim _{x \rightarrow \infty}\left[\frac{1}{x} \cdot\frac{a^{x}-1}{a-1}\right]^{1 / x}\) is 1

Step by step solution

01

Rewrite the expression

Rewrite the expression a^x as \(e^{x \cdot ln(a)}\). Using Euler's number as a base simplifies the equation and opens the door to using certain limit rules. So, the equation becomes \(\lim _{x \rightarrow \infty}\left[\frac{1}{x} \cdot\frac{e^{x \cdot ln(a)}-1}{a-1}\right]^{1 / x}\).
02

Apply l'Hopital's Rule

This equation represents an indeterminate form of 0/0 as x approaches positive infinity, so l'Hopital's rule can be applied, which states that \(\lim_{x \rightarrow a} \frac{f(x)}{g(x)} = \lim_{x \rightarrow a} \frac{f'(x)}{g'(x)}\), if the limit exists. So, apply l'Hopital's Rule, take the derivative of the numerator and denominator separately, in the given limit we get \(\lim _{x \rightarrow \infty}\frac{e^{x \cdot ln(a)}}{x}\).
03

Evaluate the limit

If we evaluate this limit as x approaches infinity, assuming a > 1, we get positive infinity over infinity, which is an indeterminate form. But since the exponential function grows faster than the linear function as x approaches infinity, the limit is infinity. If a < 1, the limit will be 0, as the exponential function decreases faster than the linear function. As a is not equal to 1 (given), the only possible solutions are 0 or infinity.
04

Apply the exponential property

Applying the property that for any real number y, the limit as x approaches infinity of (1 + y/x)^x = e^y. From previous we get \(\lim _{x \rightarrow \infty}\left(\frac{e^{x \cdot ln(a)}}{x}\right)^{1 / x}\) evaluates to \(e^0\) or 1

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Most popular questions from this chapter

Extended Mean Value Theorem In Exercises \(91-94\) , apply the Extended Mean Value Theorem to the functions \(f\) and \(g\) on the given interval. Find all values \(c\) in the interval \((a, b)\) such that $$\frac{f^{\prime}(c)}{g^{\prime}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}$$ $$ f(x)=\frac{1}{x}, \quad g(x)=x^{2}-4 \quad[1,2] $$

U-Substitution In Exercises 109 and 110 , rewrite the improper integral as a proper integral using the given u-substitution. Then use the Trapezoidal Rule with \(n=5\) to approximate the integral. $$ \int_{0}^{1} \frac{\sin x}{\sqrt{x}} d x, \quad u=\sqrt{x} $$

Improper Integral Consider the integral $$\int_{0}^{3} \frac{10}{x^{2}-2 x} d x$$ To determine the convergence or divergence of the integral, how many improper integrals must be analyzed? What must be true of each of these integrals if the given integral converges?

L'Hopital's Rule State L'Hopital's Rule.

Convergence or Divergence In Exercises \(53-62,\) use the results of Exercises \(49-52\) to determine whether the improper integral converges or diverges. $$ \int_{1}^{\infty} \frac{1}{x^{5}} d x $$
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