Let e denote the base of the natural logarithm. The value of the real number a for which the right hand limit
$\displaystyle\lim_{x\rightarrow 0^+} \frac{(1-x)^{\frac{1}{x}} - e^{-1}}{x^a}$
is equal to a nonzero real number, is_____
$\displaystyle\lim_{x\rightarrow 0^+} \frac{(1-x)^{\frac{1}{x}} - e^{-1}}{x^a}$
is equal to a nonzero real number, is_____
Correct Answer: 1
Solution and Explanation
\(\text{a non-zero real number, is}\; \underline{1}.\)
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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.