Q. 11 - Calculus

Question

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

Step-by-Step Solution

Answer

The value is $1.$

Step 1. Given information.

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

Step 2. Value of the limit.

We can write,

$$\begin{gathered} \underset{x\to \infty }{\lim }m\left(x{)}^{\frac{1}{x}}m\right(x{)}^{\frac{1}{x}}=\underset{x\to \infty }{\lim }{\left(A{x}^r\right)}^{\frac{1}{x}} \\ \underset{x\to \infty }{\lim }{\left(A{x}^r\right)}^{\frac{1}{x}}=\underset{x\to \infty }{\lim }(A{)}^{\frac{1}{x}}\underset{x\to \infty }{\lim }{\left(x^y\right)}^{\frac{1}{x}} \\ =\underset{x\to \infty }{\lim }(A{)}^{\frac{1}{x}}\underset{x\to \infty }{\lim }\left(x^{\frac{r}{x}}\right) \\ Now, a^x=e^{\ln \left(a^x\right)} \\ {x}^{\frac{r}{x}}=e^{\ln \left(x^{\frac{t}{x}}\right)}=e^{\frac{r}{x}\ln \left(x\right)} \\ \underset{x\to \infty }{\lim }\left(x^{\frac{r}{x}}\right)=\underset{x\to \infty }{\lim }{e}^{\frac{r}{x}\ln \left(x\right)} \\ =e^{\underset{x\to \infty }{\lim }\left(\frac{r}{x}\ln \left(x\right)\right)}=1 \\ Therefore, \\ \underset{x\to \infty }{\lim }{\left(A{x}^r\right)}^{\frac{1}{x}}=1 \\ \underset{x\to \infty }{\lim }m(x{)}^{\frac{1}{x}}=1 \end{gathered}$$