Question

Using the definition of limit at infinity or infinite limits, prove that

a) lim_{𝑥 → 3} 1/ (𝑥 − 3) ² = ∞ 

Answer 1

Let $C>0$ be fixed. Solve

$\left| \frac{1}{\left( x-3 \right) ^2} \right|>C\Leftrightarrow \left| x-3 \right|<\frac{1}{\sqrt{C}}$

Thus for every C>0 there exists $\delta =\frac{1}{\sqrt{C}}$ such that $\left| x-3 \right|<\delta \Rightarrow \left| \frac{1}{\left( x-3 \right) ^2} \right|>C$

By the definition

$\underset{x\rightarrow 3}{\lim}\frac{1}{\left( x-3 \right) ^2}=\infty$