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Which of the following function is continuous at $x = 0$

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$f (x) = 1, x \leq 0 : f (x) = x + \frac{1}{2}, x > 0$

$f (x) = \frac{e^{\frac{1}{x}} - x}{e^{\frac{1}{x}} + x}, x \neq 0, f (0) = 0$

$f (x) = \frac{x (1 - e^{\frac{1}{x}})}{1 + e^{\frac{1}{x}}}, x \neq 0, f (0) = 0$

$f (x) = \frac{e^{\frac{1}{x}}}{1 + e^{\frac{1}{x}}}, x \neq 0, f (0) = 0$

answer is D.

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Detailed Solution

$f (x) = \frac{x (1 - e^{\frac{1}{x}})}{1 + e^{\frac{1}{x}}} x \neq 0, f (0) = 0$

For $f (x)$ left hand limit = right hand limit = 0

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