Question:
Let $f (a) = g(a) = k$ and their nth derivatives $f^n (a), g^n (a)$ exist and are not equal for some n. Further if $\displaystyle\lim_{x \to a} \frac{f\left(a\right)g\left(x\right) -f\left(a\right) -g\left(a\right)f\left(x\right)+f\left(a\right)}{g\left(x\right)-f\left(x\right)} = 4 $
then the value of k is
Let $f (a) = g(a) = k$ and their nth derivatives $f^n (a), g^n (a)$ exist and are not equal for some n. Further if $\displaystyle\lim_{x \to a} \frac{f\left(a\right)g\left(x\right) -f\left(a\right) -g\left(a\right)f\left(x\right)+f\left(a\right)}{g\left(x\right)-f\left(x\right)} = 4 $
then the value of k is
Updated On: Jul 6, 2022
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The Correct Option is B
Solution and Explanation
$\displaystyle\lim_{x \to a} \frac{f\left(a\right)g'\left(x\right) -g\left(a\right)f'\left(x\right)}{g'\left(x\right)-f'\left(x\right)} = 4$
(By L?
$\displaystyle \lim_{x \to a} \frac{k g'\left(x\right) -kf'\left(x\right)}{g'\left(x\right)-f'\left(x\right)} =4 \therefore k = 4 $
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Concepts Used:
Continuity & Differentiability
Definition of Differentiability
f(x) is said to be differentiable at the point x = a, if the derivative f ‘(a) be at every point in its domain. It is given by
Definition of Continuity
Mathematically, a function is said to be continuous at a point x = a, if
It is implicit that if the left-hand limit (L.H.L), right-hand limit (R.H.L), and the value of the function at x=a exist and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is unspecified or does not exist, then we say that the function is discontinuous.