Consider a function defined as f x - sin x e x -1 xlog 1-x - k- x- 0 x-0 If fx is continuous at x-0- then the value of k is

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Convexity

Consider a fu...

Question

Consider a function defined as
$f (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} \frac{(sin x) (e^{x} - 1)}{x log (1 + x)}, k, \end{matrix} \begin{matrix} x \neq 0 x = 0 \end{matrix}$
If $f (x)$ is continuous at $x = 0$ , then the value of $k$ is

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f

f (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} \frac{{cos}^{2} x - {sin}^{2} x - 1}{\sqrt{x + 1} - 1} & , if x \neq 0 k & , if x = 0 \end{matrix},

x \neq 0,

x = 0

f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \frac{1 - cos x}{\begin{matrix} x k \end{matrix}} \begin{matrix} x \neq 0 x = 0 \end{matrix} ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

x = 0

k

f (x) = \{\begin{matrix} \frac{1 - \cos 2 x}{2 x^{2}}, & x < 0 \\ k, & x = 0 \\ \frac{x}{|x|}, & x > 0 \end{matrix}

f

f (x) = \frac{1}{x} - \frac{k - 1}{e^{2 x} - 1}, x \neq 0

x = 0

f

f (x) = \frac{1}{x} - \frac{k - 1}{e^{2 x} - 1}

x \neq 0

x = 0

(k, f (0))

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