Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{e^x - 1}{\log_e (1 + 2x)}, & \text{ if }x eq 0 \\ 7 , & \text{ if } x = 0\end{cases}\]

Given: \[f\left( x \right) = \begin{cases}\frac{e^x - 1}{\log_e (1 + 2x)}, & \text{ if }x eq 0 \\ 7 , & \text{ if } x = 0\end{cases}\] We have \[\lim_{x \to 0} f\left( x \right) = \lim_{x \to 0} \frac{e^x - 1}{\log_e \left( 1 + 2x \right)} = \lim_{x \to 0} \frac{\left( \frac{e^x - 1}{x} \right)}{\left( \frac{2 \log_e \left( 1 + 2x \right)}{2x} \right)} = \frac{1}{2} \times \frac{\lim_{x \to 0} \left( \frac{e^x - 1}{x} \right)}{\lim_{x \to 0} \left( \frac{\log_e \left( 1 + 2x \right)}{2x} \right)} = \frac{1}{2}\] It is given that \[f\left( 0 \right) = 7\] ⇒ \[\lim_{x \to 0} f\left( x \right) eq f\left( 0 \right)\] Hence, the given function is discontinuous at x = 0 and continuous elsewhere.

Find the Points of Discontinuity, If Any, of the Following Functions: F ( X ) = { E X − 1 Log E ( 1 + 2 X ) , If X ≠ 0 7 , If X = 0 - Mathematics

प्रश्न

Find the points of discontinuity, if any, of the following functions:

f (x) = {\begin{cases} \frac{e^{x} - 1}{\log_{e} (1 + 2 x)}, & if x \neq 0 \\ 7, & if x = 0 \end{cases}

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उत्तर

Given:

f (x) = {\begin{cases} \frac{e^{x} - 1}{\log_{e} (1 + 2 x)}, & if x \neq 0 \\ 7, & if x = 0 \end{cases}

We have

$lim_{x \to 0} f (x) = lim_{x \to 0} \frac{e^{x} - 1}{\log_{e} (1 + 2 x)} = lim_{x \to 0} \frac{(\frac{e^{x} - 1}{x})}{(\frac{2 \log_{e} (1 + 2 x)}{2 x})} = \frac{1}{2} \times \frac{lim_{x \to 0} (\frac{e^{x} - 1}{x})}{lim_{x \to 0} (\frac{\log_{e} (1 + 2 x)}{2 x})} = \frac{1}{2}$

It is given that

f (0) = 7

⇒

lim_{x \to 0} f (x) \neq f (0)

Hence, the given function is discontinuous at x = 0 and continuous elsewhere.

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Q 3.07 Q 3.06 Q 3.08

अध्याय 9: Continuity - Exercise 9.2 [पृष्ठ ३४]

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अध्याय 9 Continuity
Exercise 9.2 | Q 3.07 | पृष्ठ ३४

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Find the Points of Discontinuity, If Any, of the Following Functions: F ( X ) = { E X − 1 Log E ( 1 + 2 X ) , If X ≠ 0 7 , If X = 0 - Mathematics

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