Limit of a Function

==2024-11-14

Theory

For a function $y=f(x)$, the limit of the function at $x_o$, $L$ is the value of the function when $x$ approaches $x_o$.

${\lim }_{x\to x_0}f(x)=L$ if For each $ϵ\text{ (small number(y-axis)) }>0$, there exist a $\delta \text{ (another small number(x-axis))}$ such that, $|x-x_o|<\delta ⟹|f(x)-L|<ϵ$.

This means the difference between $x$ and $x_o$ is very small and so is the output($f(x)$) and the limit($L$), even smaller than the very small numbers $ϵ$ and $\delta$ respectively.

In limit we approach $x_o$ from minimum possible distance. So, $\delta$ is taken min{set of all possible values of $\delta$}.

Important

To prove limit of a function, see the values for $f(x)$, $x_o$ ,and $L$

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Example

1. $f(x)=\{\begin{array}{ll}{x}^2& \text{if }x\ne 1\\ 2& \text{if }x=1\end{array}⟹{\lim }_{x\to 1}f(x)=1$

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PTR

1. ${\lim }_{x\to x_0}f(x)=L$ $⟺$ ($|x-x_o|<\delta ⟹|f(x)-L|<ϵ$)