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Chapter 10: Problem 7

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow 0} \frac{e^{x}-1}{x}$$

Short Answer

Expert verified
Answer: The limit of the expression \(\frac{e^{x}-1}{x}\) as x approaches 0 is 1.

Step by step solution

01

Rewrite the expression using Taylor series for exponential function

We start by replacing the \(e^{x}\) in the expression with its Taylor series $$\lim _{x \rightarrow 0} \frac{(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots) - 1}{x}$$
02

Simplify the expression

Next, we simplify the expression by cancelling out the 1 from the numerator: $$\lim _{x \rightarrow 0} \frac{x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots}{x}$$
03

Divide the expression by x

Now, we see that all terms have x as a factor, so we divide each term by x: $$\lim _{x \rightarrow 0} \left(1 + \frac{x}{2!} + \frac{x^2}{3!} + \dots\right)$$
04

Compute the limit as x approaches 0

Finally, we compute the limit as x approaches 0. All terms with x in the denominator go to 0: $$\lim _{x \rightarrow 0} \left(1 + \frac{x}{2!} + \frac{x^2}{3!} + \dots\right) = 1$$ Thus, the value of the given limit is 1.

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Most popular questions from this chapter

Use composition of series to find the first three terms of the Maclaurin series for the following functions. a. \(e^{\sin x}\) b. \(e^{\tan x}\) c. \(\sqrt{1+\sin ^{2} x}\)

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. $$f(x)=b^{x}, \text { for } b > 0, b \neq 1$$

Let \(f(x)=\left(e^{x}-1\right) / x,\) for \(x \neq 0\) and \(f(0)=1 .\) Use the Taylor series for \(f\) and \(f^{\prime}\) about 0 to evaluate \(f^{\prime}(2)\) to find the value of \(\sum_{k=1}^{\infty} \frac{k 2^{k-1}}{(k+1) !}\)

Consider the following function and its power series: $$ f(x)=\frac{1}{(1-x)^{2}}=\sum_{k=1}^{\infty} k x^{k-1}, \quad \text { for }-1

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The Taylor polynomials for \(f(x)=e^{-2 x}\) centered at 0 consist of even powers only. b. For \(f(x)=x^{5}-1,\) the Taylor polynomial of order 10 centered at \(x=0\) is \(f\) itself. c. The \(n\) th-order Taylor polynomial for \(f(x)=\sqrt{1+x^{2}}\) centered at 0 consists of even powers of \(x\) only.
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