Determine \begin{equation}\lim_{x\to 2}\,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}

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

Determine \begin{equation}\lim_{x\to 2}\,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: 

10

SAMPLE SOLUTION

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Solution: 

First, note that when <Sign in to see all the formulas> then the function reduces to $5x$ and we know that

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While this function may not be continuous at $x=1$, it is continuous at $x=2$ which is where we want to find the limit. We therefore assert that

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and attempt to prove it.

Now choose an arbitrary <Sign in to see all the formulas> and <Sign in to see all the formulas> where <Sign in to see all the formulas> is the <Sign in to see all the formulas> used to prove

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Assume that we are at some value of $x$ such that <Sign in to see all the formulas>. Since <Sign in to see all the formulas> in this range we know that <Sign in to see all the formulas> and by definition

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