Question

The function $f(x)=(2)^{\frac{1}{x}}$ is not continuous at}

• A. $x=0$
• B. $x=1$
• C. $x=-1$
• D. $x=2$

Hint:

Always look for denominators: if $x$ is in the denominator, $x=0$ is usually a point of discontinuity.

Answer 1

The Correct Option is A

Step 1: Concept A function is discontinuous at a point if the limit from the left, the limit from the right, or the function value at that point does not exist or are not equal.

Step 2: Meaning For $f(x) = 2^{1/x}$, the exponent becomes undefined when the denominator is zero.

Step 3: Analysis At $x=0$, the right-hand limit ($x \to 0^+$) is $2^{\infty} = \infty$, while the left-hand limit ($x \to 0^-$) is $2^{-\infty} = 0$.

Step 4: Conclusion Since the limits are unequal and the function is undefined at $x=0$, it is discontinuous there. Final Answer: (A)