Determine \begin{equation}\lim_{x\to 3}\,f(x)\end{equation} where \begin{equation}f(x)=\left\{\begin{array}{lr}x^2,&x\lt3\\10,&x=3\\2x+3,&x\gt 3\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 3}\,f(x)\end{equation} where \begin{equation}f(x)=\left\{\begin{array}{lr}x^2,&x\lt3\\10,&x=3\\2x+3,&x\gt 3\end{array}\right.\end{equation}

Answer: 

9

SAMPLE SOLUTION

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Solution: 

Recall that the function limit does not have to equal the function value at the point you are trying to determine the limit. In other words, when

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then $f(a)$ does not have to equal $L$.

In this particular problem, the function $x^2$ and $2x+3$ both are $9$ when $x=3$. So on both sides of $x=3$ the two functions approach $9$. Therefore, we assert that

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and will attempt to prove this. Because we have two different functions on the left and right side of $x=3$, we will use our one-sided limits to prove that the limit from the left and the limit from the right are both equal to $9$.

Note that <Sign in to see all the formulas> and not our function limit $9$.

First, both $x^2$ and $2x+3$ have function limits proven here and here, respectively, so that

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Therefore, we know that the left and right-sided limits

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and

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exist. Therefore,

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Because the right and left-sided limits exist and are equal we know that

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