Approach #1: Create equivalent fractions
Key property: We can create equivalent fractions by taking a fraction and multiplying the numerator and denominator by the same value

When I scan the answer choices, I see that answer choice E looks very similar to the given fraction $\frac{1}{\frac{1}{x}+1}$
Notice that, if we take $\frac{1}{\frac{1}{x}+1}$.....
.... And multiply numerator and denominator by 2, we get the equivalent fraction $\frac{2}{\frac{2}{x}+2}$

In other words $\frac{1}{\frac{1}{x}+1} = \frac{2}{\frac{2}{x}+2}$

So, if $\frac{1}{\frac{1}{x}+1}=k$, then it must also be true that $\frac{2}{\frac{2}{x}+2} = k$

Answer: E

Approach #2: Test a value of x

If $x = 1$, then we get: $\frac{1}{\frac{1}{x}+1} = \frac{1}{\frac{1}{1}+1} = \frac{1}{2}$
So, it must be the case that the correct answer also evaluates to $\frac{1}{2}$, when $x = 1$

We'll now plug $x = 1$ into each answer choice to see which one evaluates to $\frac{1}{2}$

A. $\frac{1}{\frac{2}{1}+2}= \frac{1}{4}$. No good. We need $\frac{1}{2}$. Eliminate

B. $\frac{2}{\frac{1}{2(2)}+1}= \frac{8}{5}$. No good. We need $\frac{1}{2}$. Eliminate

C. $\frac{2}{\frac{2}{1}+1}= \frac{2}{3}$. No good. We need $\frac{1}{2}$. Eliminate

D. $\frac{2}{\frac{1}{2(2)}+\frac{1}{2}}= \frac{8}{3}$. No good. We need $\frac{1}{2}$. Eliminate

E. $\frac{2}{\frac{2}{1}+2}= \frac{1}{2}$ PERFECT

Answer: E