Question

it is a fact that the function f(x)=(\frac{x + 3}{x})^x has a limiting value. use a table of values to estimate the limiting value. (round your answer to two decimal places.)

Explanation:

Step1: Recall the limit formula

We know that $\lim_{x
ightarrow\infty}(1 + \frac{a}{x})^x=e^a$. Rewrite $f(x)=(\frac{x + 3}{x})^x$ as $f(x)=(1+\frac{3}{x})^x$.

Step2: Apply the limit rule

As $x
ightarrow\infty$, using the formula $\lim_{x
ightarrow\infty}(1+\frac{a}{x})^x = e^a$ with $a = 3$, we get $\lim_{x
ightarrow\infty}(1+\frac{3}{x})^x=e^3$.

Step3: Calculate the value

We know that $e\approx2.71828$, so $e^3\approx20.09$.

Answer:

$20.09$