Evaluate the continued fraction:

$x=3+\frac{1}{2+\frac{1}{5+\frac{1}{3}}}$

The correct option is $\frac{121}{35}$.

To evaluate a finite continued fraction, we start simplifying from the innermost (bottom-most) denominator and work our way upward step-by-step.

Step 1: Simplify the innermost denominator.

The bottom-most fraction is $5+\frac{1}{3}$.

Converting 5 into a fraction with a denominator of 3 gives:

$5+\frac{1}{3}=\frac{15}{3}+\frac{1}{3}=\frac{16}{3}$

Step 2: Take the reciprocal of this value.

Substituting this back into the continued fraction, the middle section becomes:

$\frac{1}{5+\frac{1}{3}}=\frac{1}{16/3}=\frac{3}{16}$

Step 3: Simplify the next level up.

Now we evaluate $2+\frac{3}{16}$:

$2+\frac{3}{16}=\frac{32}{16}+\frac{3}{16}=\frac{35}{16}$

Step 4: Take the reciprocal of this result.

The fraction term under 3 becomes:

$\frac{1}{35/16}=\frac{16}{35}$

Step 5: Perform the final addition.

Finally, we add 3 to this fraction:

$x=3+\frac{16}{35}=\frac{3\times 35}{35}+\frac{16}{35}=\frac{105}{35}+\frac{16}{35}=\frac{121}{35}$

Thus, the value of the continued fraction is $\frac{121}{35}$.