Question:
If the function \[ f(x) =
\begin{cases}
[ tan (\frac {\pi}{4}+x)]^{1/x} & \quad for\, x \neq 0\\
K \,\,\,\,\,\,\,\,\,\text{if } x =0
\end{cases}
\]
is continuous at $x = 0$, then $K = ?$
If the function \[ f(x) =
\begin{cases}
[ tan (\frac {\pi}{4}+x)]^{1/x} & \quad for\, x \neq 0\\
K \,\,\,\,\,\,\,\,\,\text{if } x =0
\end{cases}
\]
is continuous at $x = 0$, then $K = ?$
Updated On: May 8, 2024
- $e$
- $e^{-1}$
- $e^2$
- $e^{-2}$
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The Correct Option is C
Solution and Explanation
Answer (c) $e^2$
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Concepts Used:
Continuity
A function is said to be continuous at a point x = a, if
limx→a
f(x) Exists, and
limx→a
f(x) = f(a)
It implies that if the left hand limit (L.H.L), right hand limit (R.H.L) and the value of the function at x=a exists and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is undefined or does not exist, then we say that the function is discontinuous.
Conditions for continuity of a function: For any function to be continuous, it must meet the following conditions:
- The function f(x) specified at x = a, is continuous only if f(a) belongs to real number.
- The limit of the function as x approaches a, exists.
- The limit of the function as x approaches a, must be equal to the function value at x = a.