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While finding the limit of the function $g(x)$, both left and right hand side limits are equal at x=a but at the point $g(a)$ is different. In this case, whether the limit exists?

For example,

$$g(x)= \begin{cases}2x+2 & \text{ if } x<2,\\ 8 &\text{ if } x=2 ,\\ 4x-2&\text{ if } x>2.\end{cases}$$

Here, both left and right side limit exists and equal to $6$, however, $g(2)=8$. In this case, what is $\displaystyle\lim_{x\rightarrow 2} g(x) = ?$. Is it does not exist or any specific value?

Aruha
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1 Answers1

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The value of the function at $x=a$ doesn’t matter for the definition of limit. Since both right and left limit exist then by the definition we say that the limit of the function at $x=2$ is equal to $6$.

What is true is that the function is not continuous at that point but for the definition of limit, the function is not even requested to by defined at that point (e.g. $\sin x/x$ at $x=0$).

user
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