Lecture 8: One Sided Limits

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

\rightarrow Two sides limits


Functions can have one sided limit if approach from one side

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

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


Rule

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

limxcf(x)=L\lim_{x \rightarrow c} f(x) = L

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

Rule

Also remember this side rule to help solve problem

limθ0sinθθ=1\lim_{\theta \rightarrow 0} \frac{\sin \theta}{\theta} = 1, where θ\theta can be replaced by xx


Example 1

Does the limit exist for limx3x3x3\lim_{x \rightarrow 3} \frac{|x-3|}{x-3} ?

RHLlimx3+x3x3=limx3+x3x3=x3x3=1RHL \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 

LHLlimx3x3x3=limx3x3x3=(x3)x3=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

11\Rightarrow 1 \neq 1 \therefore  Limit DNE


EXAMPLE 2

limx0sin3x4x=limx0sin3x4/33x=34limx0sin3x3x=341=34\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}



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