Answer
False.
Work Step by Step
Consider
$f(x)=\displaystyle \frac{1}{x^{2}},\qquad g(x)=\frac{1}{x^{4}}$
(Both have even exponents to force them to be positive)
$\displaystyle \lim_{x\rightarrow 0}f(x)=\infty,\qquad \displaystyle \lim_{x\rightarrow 0}g(x)=\infty$
$\displaystyle \lim_{x\rightarrow 0}[f(x)-g(x)]=\lim_{x\rightarrow 0}(\frac{1}{x^{2}}-\frac{1}{x^{4}})$
$=\displaystyle \lim_{x\rightarrow 0}(\frac{x^{2}-1}{x^{4}})=\lim_{x\rightarrow 0}(\frac{x^{2}-1}{x^{4}}\cdot\frac{\frac{1}{x^{2}}}{\frac{1}{x^{2}}})$
$=\displaystyle \lim_{x\rightarrow 0}\frac{1-\frac{1}{x^{2}}}{x^{2}}$
The numerator approaches 1, the denominator approaches zero...
this limit is not 0,
$\displaystyle \lim_{x\rightarrow 0}[f(x)-g(x)]=\infty$
so the statement is false.
