Limits…

This is a Spivak's Problem, and we can read the solution at the end of the book with epsilon-delta definition. Suppose that $\lim _{ x\rightarrow 0 }{ f\left( x \right)}=L$ Define $g\left(x\right)=f\left(x-a\right)$ Note that $\lim_{x\rightarrow a}{g\left(x\right)}=L\Longleftrightarrow\lim_{h\rightarrow 0}{g\left(a+h\right)}=L$ Therefore: $\lim_{h\rightarrow 0}{g\left(a+h\right)}=\lim_{h\rightarrow 0}{f\left(a+h-a\right)}=\lim_{h\rightarrow 0}{f\left(h\right)}$ But $\lim_{h\rightarrow 0}{f\left(h\right)}$ exists and is $L$ , we can conclude that $\lim_{x\rightarrow a}{g\left(x\right)}=L$

If the solution is wrong, comment.