Question

question 1. version 1*/1. score: 0/4 consider $f(x)=(1 + x)^{\frac{-2}{x}}$
state your answers to 5 decimal places. looking at the values, we can estimate that $lim_{x
ightarrow0}f(x)=$ score: n/a/0.571 n/a/0.571 n/a/0.571 n/a/0.571

Explanation:

Step1: Substitute $x = - 0.01$ into $f(x)=(1 + x)^{\frac{-2}{x}}$

Let $x=-0.01$, then $f(-0.01)=(1-0.01)^{\frac{-2}{-0.01}}=(0.99)^{200}$. Using a calculator, $(0.99)^{200}\approx0.13406$.

Step2: Substitute $x = 0.01$ into $f(x)=(1 + x)^{\frac{-2}{x}}$

Let $x = 0.01$, then $f(0.01)=(1 + 0.01)^{\frac{-2}{0.01}}=(1.01)^{- 200}$. Using a calculator, $(1.01)^{-200}\approx0.13262$.

Step3: Substitute $x=-0.001$ into $f(x)=(1 + x)^{\frac{-2}{x}}$

Let $x=-0.001$, then $f(-0.001)=(1-0.001)^{\frac{-2}{-0.001}}=(0.999)^{2000}$. Using a calculator, $(0.999)^{2000}\approx0.13520$.

Step4: Substitute $x = 0.001$ into $f(x)=(1 + x)^{\frac{-2}{x}}$

Let $x = 0.001$, then $f(0.001)=(1 + 0.001)^{\frac{-2}{0.001}}=(1.001)^{-2000}$. Using a calculator, $(1.001)^{-2000}\approx0.13440$.

Step5: Substitute $x=-0.0001$ into $f(x)=(1 + x)^{\frac{-2}{x}}$

Let $x=-0.0001$, then $f(-0.0001)=(1-0.0001)^{\frac{-2}{-0.0001}}=(0.9999)^{20000}$. Using a calculator, $(0.9999)^{20000}\approx0.13534$.

Step6: Substitute $x = 0.0001$ into $f(x)=(1 + x)^{\frac{-2}{x}}$

Let $x = 0.0001$, then $f(0.0001)=(1 + 0.0001)^{\frac{-2}{0.0001}}=(1.0001)^{-20000}$. Using a calculator, $(1.0001)^{-20000}\approx0.13454$.

Step7: Estimate the limit

As $x$ approaches $0$ from both the left - hand side and the right - hand side, the values of $f(x)$ seem to be approaching approximately $0.135$.

Answer:

$x$	$f(x)$	$x$	$f(x)$
$-0.001$	$0.13520$	$0.001$	$0.13440$
$-0.0001$	$0.13534$	$0.0001$	$0.13454$

$\lim_{x
ightarrow0}f(x)\approx0.135$