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Question

# For the following exercise,With the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as $x$ approaches $a$. If the function has a limit as $x$ approaches $a$, state it. If not, discuss why there is no limit.$f(x)=\left\{\begin{array}{ll}\frac{1}{x}-3, & \text { if } x \leq 2 \\ x^3+1, & \text { if } x>2\end{array} a=2\right.$

Solution

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Step 1
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In order to solve this, first, we need to check what we get for

$\lim_{x\to 2^-}f(x).$

We can see from the graph below that as we take values of $x$ sufficiently close to $2$, such that $x<2$ and $x\neq2$, the output values approach to $-2.5$.

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