Chapter 7: Q. 11 (page 639)

Let $m (x) = A x^{r}$ be a power function. Evaluate $\lim_{x \to \infty} m (x)^{1 / x}$ .

Short Answer

Expert verified

The value is $1 .$

Step by step solution

Step 1. Given information.

We are given $\lim_{x \to \infty} m (x)^{1 / x}$ .

Step 2. Value of the limit.

We can write,

$\lim_{x \to \infty} m (x)^{\frac{1}{x}} m (x)^{\frac{1}{x}} = \lim_{x \to \infty} {(A x^{r})}^{\frac{1}{x}} \lim_{x \to \infty} {(A x^{r})}^{\frac{1}{x}} = \lim_{x \to \infty} (A)^{\frac{1}{x}} \lim_{x \to \infty} {(x^{y})}^{\frac{1}{x}} = \lim_{x \to \infty} (A)^{\frac{1}{x}} \lim_{x \to \infty} (x^{\frac{r}{x}}) N o w, a^{x} = e^{\ln (a^{x})} x^{\frac{r}{x}} = e^{\ln (x^{\frac{t}{x}})} = e^{\frac{r}{x} \ln (x)} \lim_{x \to \infty} (x^{\frac{r}{x}}) = \lim_{x \to \infty} e^{\frac{r}{x} \ln (x)} = e^{\lim_{x \to \infty} (\frac{r}{x} \ln (x))} = 1 T h e r e f o r e, \lim_{x \to \infty} {(A x^{r})}^{\frac{1}{x}} = 1 \lim_{x \to \infty} m (x)^{\frac{1}{x}} = 1$

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Let $m (x) = A x^{r}$ be a power function. Evaluate $\lim_{x \to \infty} m (x)^{1 / x}$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Value of the limit.

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Let m(x)=Axrbe a power function. Evaluatelimx→∞m(x)1/x.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Value of the limit.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

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Statistics

Study anywhere. Anytime. Across all devices.

Let $m (x) = A x^{r}$ be a power function. Evaluate $\lim_{x \to \infty} m (x)^{1 / x}$ .