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Question
Find the points of discontinuity, if any, of the following functions:
Solution
Given:
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
Hence, the given function is discontinuous at x = 0 and continuous elsewhere.
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