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What do mathematicians mean by $\lim _{x \rightarrow a} f(x)=\mathbf{L}$
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Let $y=f(x)$ be a function. Suppose that $a$ and $\mathbf{L}$ are numbers such that:

• whenever $\boldsymbol{x}$ is close to $\boldsymbol{a}$ but not equal to $\boldsymbol{a}, \boldsymbol{f}(\boldsymbol{x})$ is close to $\mathbf{L} ;$
• as $\boldsymbol{x}$ gets closer and closer to $\boldsymbol{a}$ but not equal to $\boldsymbol{a}, \boldsymbol{f}(\boldsymbol{x})$ gets closer and closer to $\mathbf{L}$; and
• suppose that $f(x)$ can be made as close as we want to $\mathbf{L}$ by making $x$ close to $a$ but not equal to $a$.

Then we say that the limit of $f(x)$ as $x$ approaches a is $L$ and we write $\lim _{x \rightarrow a} f(x)=\mathbf{L}$

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