5. Case 3 - Limit to Infinity

Case 3

Find the $N$ & $\epsilon$ that satisfies

$$\lim_{x \to \pm\infty}\dfrac{1}{x} = 0$$

Answer

For limits at infinity, we use the $\epsilon-N$ definition

For any $\epsilon > 0$, there exists $N > 0$ such that

$$|x| > N \quad \to \quad |\dfrac{1}{x} - 0| < \epsilon$$

Step 1: Write the inequaltiy in terms of x

$$|\dfrac{1}{x}| < \epsilon$$

Step 2: Solve for $|x|$

$$|x| > \dfrac{1}{\epsilon}$$

Step 3: Choose $N$

$$N = \dfrac{1}{\epsilon}$$