Write the value of _ x f(x)-f(c) x-c

Question

Write the value of $\lim_\limits{\text{x}\rightarrow\text{c}}\frac{\text{f}(\text{x})-\text{f}(\text{c})}{\text{x}-\text{c}}$

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Answer

Since we know that,$\frac{\text{d}}{\text{dx}}(\text{f}(\text{x}))=\lim_\limits{\text{h}\rightarrow0}\frac{\text{f}(\text{x}+\text{h}-\text{f}(\text{x}))}{\text{h}}$
Let,
$\text{h}=\text{x}-\text{c}$ and $\text{f}(\text{x})=\text{c}$
If $\text{h}\rightarrow0$ then $\text{x}\rightarrow\text{c}$
Therefore,
$\frac{\text{d}}{\text{dx}}(\text{c})=\lim_\limits{\text{x}\rightarrow\text{c}}\frac{\text{f}(\text{c}+\text{x}-\text{c})-\text{f}(\text{c})}{\text{x}-\text{c}}$
$=\lim_\limits{\text{x}\rightarrow\text{c}}\frac{\text{f}(\text{x})-\text{f}(\text{c})}{\text{x}-\text{c}}$
$=\text{f}'(\text{c})$

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