# Lecture 10: Limits Involving Infinity & Asymptotes of a Graph

### Limits

﻿$\lim f(x) = L (finite)$﻿

﻿$x\rightarrow \pm\infty$﻿

Remember ﻿$\rightarrow$﻿ ﻿$\lim \frac{1}{x^p} = 0$﻿ where ﻿$p = \pm$﻿

﻿$x\rightarrow \pm\infty$﻿

﻿$\lim k=k$﻿ where K = constant

﻿$x\rightarrow \pm\infty$﻿

### Horizontal Asymptote

The line y=L is H.A of the curve ﻿$y=f(x)$﻿ if ﻿$\rightarrow$﻿

﻿$\lim f(x) = L$﻿ or ﻿$\lim f(x) = L$﻿

﻿$x→+∞$﻿ ﻿$x→-∞$﻿

Remember this!

1. ﻿$\lim \frac{x-3}{x2} + x-7 = 0$﻿

﻿$x→∞$﻿

2. ﻿$\lim \frac{ x2 +5x-7}{x-3} = ∞$﻿ no. HA

﻿$x→∞$﻿

3. ﻿$\lim \frac{x2 +5x-7}{4x2 + 3} = \frac{1}{4}$﻿ therefore HA @ ﻿$y= \frac{1}{4}$﻿

﻿$x→∞$﻿

4. ﻿$\lim e^x = 0$﻿ & ﻿$\lim e^{-x} = ∞$﻿

﻿$x→∞$﻿

5. ﻿$\lim e^x = ∞$﻿ & ﻿$\lim e^{-x} = 0$﻿

﻿$x→∞$﻿

6. ﻿$\lim \frac{1}{x2}= ∞$﻿ therefore ﻿$\lim DNE$﻿ & ﻿$\lim \frac{1}{x} = -∞$﻿ ,﻿$∞$﻿ therefore ﻿$\lim DNE$﻿

﻿$x→0$﻿ ﻿$x→0$﻿

### Vertical Asymptote

The line ﻿$x=a$﻿ is a VA of the curve ﻿$y= f(x)$﻿ if

﻿$\lim f(x) = \pm∞$﻿ or ﻿$\lim = \pm∞$﻿

﻿$x→a+$﻿ ﻿$x→-∞$﻿