(Solved): EXAMPLE 5 Use this definition to prove the following limit. \[ \lim _{x \rightarrow 0} \frac{1}{x^ ...

EXAMPLE 5 Use this definition to prove the following limit. \[ \lim _{x \rightarrow 0} \frac{1}{x^{4}}=\infty \] SOLUTION Let \( M \) be a given positive number. We want to find a number \( \delta \) such that if \( \quad 0<|x|<\delta \quad \) then \( \quad 1 / x^{4}>M \). But \( \quad \frac{1}{x^{4}}>M \quad \Leftrightarrow \quad<\frac{1}{M} \Leftrightarrow|x|< \) So if we choose \( \delta= \) and \( 0<|x|<\delta= \) , then \( 1 / x^{4}>M \). This shows that \( 1 / x^{4} \rightarrow \infty \) as \( x \rightarrow 0 \)