Determine lim

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Lesson Parent: 
Function Limits IV: The Gory Details Of One-Sided Limits
Problem: 

Determine \begin{equation}\lim_{x\to 1}\,f(x)\quad\mbox{where}\quad f(x)=\left\{\begin{array}{lr}x^2,&x\leq 1\\5x,&x\gt 1\end{array}\right.\end{equation}

Answer: 

The function limit does not exist since the left-side and right-side limits are not equal.

SAMPLE SOLUTION

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Solution: 

We know that

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and therefore know that

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This means that for f(x) that

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Since the limit from the left and the limit from the right do not equal,

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does not exist.