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If Lim X → C F ( X ) − F ( C ) X − C Exists Finitely, Write the Value of Lim X → C F ( X ) - Mathematics

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प्रश्न

If  \[\lim_{x \to c} \frac{f\left( x \right) - f\left( c \right)}{x - c}\]  exists finitely, write the value of  \[\lim_{x \to c} f\left( x \right)\]

संक्षेप में उत्तर
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उत्तर

 Given:   

\[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\]
\[\lim_{x \to c} \frac{f(x) - f(c)}{x - c} = f'(c)\]

Now,

\[\lim_{x \to c} f(x) = \lim_{x \to c} \left[ \left\{ \frac{f(x) - f(c)}{x - c} \right\} (x - c) + f(c) \right]\]
\[ = \lim_{x \to c} \left[ \left\{ \frac{f(x) - f(c)}{x - c} \right\} (x - c) \right] + f(c)\]
\[ = \lim_{x \to c} \left\{ \frac{f(x) - f(c)}{x - c} \right\} \lim_{x \to c} (x - c) + f(c)\]
\[ = f'(c) \times 0 + f(c)\]
\[ = f(c)\]

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अध्याय 10: Differentiability - Exercise 10.3 [पृष्ठ १७]

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आरडी शर्मा Mathematics [English] Class 12
अध्याय 10 Differentiability
Exercise 10.3 | Q 11 | पृष्ठ १७

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