Sorry to disappoint, but none of these answers are correct. All three limits are equal to 7.
Compute the one-sided limits:
[tex]\displaystyle\lim_{x\to2^-}h(x)=\lim_{x\to2}(x^2-x+5)=2^2-2+5=7[/tex]
[tex]\displaystyle\lim_{x\to2^+}h(x)=\lim_{x\to2}(x^3-1)=2^3-1=7[/tex]
Both the one-sided limits agree, so the two-sided limit has the same value,
[tex]\displaystyle\lim_{x\to2}h(x)=7[/tex]
