Let the function f ( x ) = { log e ( 1 + 5 x ) - log e ( 1 + x ) x , if x 0 10 , if x = 0 be continuous at x = 0 . Then absolute value of is equal to

Question

Let the function

$f (x) =$ ${\begin{matrix} \frac{\log_{e} (1 + 5 x) - \log_{e} (1 + α x)}{x}, & if x \neq 0 \\ 10, & if x = 0 \end{matrix}$

be continuous at $x = 0$ . Then absolute value of $' α'$ is equal to

Smart Achievers · Accepted Answer

(4)

$Given function is$

$\lim_{x \to 0} \frac{(5 x + \dots) - \ln (1 + α x)}{x} = 0$

$f (x) =$ ${\begin{matrix} \frac{\ln (1 + 5 x) - \ln (1 + α x)}{x}, & x \neq 0 \\ 10, & x = 0 \end{matrix}$

$Applying expansion of \ln (1 + x)$

$\lim_{x \to 0} (5 - α) = 10$

$5 - α = 10$

$\Rightarrow α = - 5$

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