Question

Consider the function $f(x)=xe^{-\frac{2}{x}}$. Which one of the following is not true?

• A. $f(x)$ is continuous at $x=1$
• B. $f(x)$ is continuous at $x=-1$
• C. $f(x)$ is continuous at $x=2$
• D. $f(x)$ is continuous at $x=-2$
• E. $f(x)$ is continuous at $x=0$

Hint:

Logic Tip: In multiple-choice questions asking "which is not true" regarding continuity, always look for the point that makes a denominator zero or falls outside the domain of a logarithm/root. That point is an automatic discontinuity!

Answer 1

Answer: (E)

Concept:
Calculus - Continuity and Domain of Exponential Functions.
A function is continuous at all points in its domain. If a function is undefined at a point, it has a point of discontinuity there.
Step 1: Analyze the components of the function.
The given function is $f(x) = x \cdot e^{-\frac{2}{x}}$. It is the product of two functions:

• A polynomial function $g(x) = x$, which is continuous everywhere on $\mathbb{R}$.
• A composite exponential function $h(x) = e^{-\frac{2}{x}}$.

Step 2: Determine the domain restrictions.
The composite function $e^{-\frac{2}{x}}$ involves division by $x$ in its exponent. Division by zero is undefined in mathematics. Therefore, the exponent $-\frac{2}{x}$ is undefined when $x = 0$.
Step 3: Evaluate the continuity at non-zero points.
For any non-zero real number (e.g., $x = 1, -1, 2, -2$), the function consists of continuous operations (multiplication and exponentiation) on defined values. Thus, $f(x)$ is continuous at all $x \ne 0$, confirming that options A, B, C, and D are true statements.
Step 4: Evaluate the behavior at $x = 0$.
Since the function $f(x)$ is explicitly undefined at $x = 0$, the condition $f(0) = \lim_{x\rightarrow0} f(x)$ cannot possibly be met. Therefore, the function is discontinuous at $x = 0$. The statement "$f(x)$ is continuous at $x=0$" is NOT true.