## Calculus 8th Edition

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.