# Lecture 8: One Sided Limits

To have a limit ﻿$L$﻿ as ﻿$x$﻿ approaches ﻿$c$﻿, the function must be defined on both sides of c and the ﻿$f(x)$﻿ values must approach ﻿$L$﻿

﻿$\rightarrow$﻿ Two sides limits

Functions can have one sided limit if approach from one side

﻿$\lim_{x \rightarrow c^+} f(x) = L$﻿ ﻿$\Rightarrow$﻿ A right hand limit

﻿$\lim_{x \rightarrow c^-} f(x) = L$﻿ ﻿$\Rightarrow$﻿ A left hand limit

#### Rule

The limit exits if a function ﻿$f(x)$﻿ has a limit as ﻿$x$﻿ approaches ﻿$c$﻿. If it has left handed and right handed limits and both are equal

﻿$\lim_{x \rightarrow c} f(x) = L$﻿

If ﻿$\lim_{x \rightarrow c^-} f(x) = L=\lim_{x \rightarrow c^+} f(x)$﻿

#### Rule

Also remember this side rule to help solve problem

﻿$\lim_{\theta \rightarrow 0} \frac{\sin \theta}{\theta} = 1$﻿, where ﻿$\theta$﻿ can be replaced by ﻿$x$﻿

#### Example 1

Does the limit exist for ﻿$\lim_{x \rightarrow 3} \frac{|x-3|}{x-3}$﻿?

﻿$RHL \rightarrow \lim_{x \rightarrow 3^+} \frac{|x-3|}{x-3} = \lim_{x \rightarrow 3^+} \frac{x-3}{x-3} = \frac {x-3}{x-3}=1$﻿

﻿$LHL \rightarrow \lim_{x \rightarrow 3^-} \frac{|x-3|}{x-3} = \lim_{x \rightarrow 3^-} \frac{-x-3}{x-3} = \frac {-(x-3)}{x-3}=-1$﻿

﻿$\Rightarrow 1 \neq 1 \therefore$﻿ Limit DNE

#### EXAMPLE 2

﻿$\lim_{x \rightarrow 0} \frac{\sin 3x}{4x} = \lim_{x \rightarrow 0} \frac{\sin 3x}{4/3 \cdot 3x}= \frac{3}{4} \lim_{x \rightarrow 0} \frac{\sin 3x}{3x} = \frac{3}{4} \cdot 1 = \frac{3}{4}$﻿