Question:
Let \( f(x) = \frac{x + |x|(1 + x)}{x} \sin \left( \frac{1}{x} \right) \), for \( x \neq 0 \). Write \( L = \lim_{x \to 0^-} f(x) \) and \( R = \lim_{x \to 0^+} f(x) \). Then which one of the following is TRUE?
Let \( f(x) = \frac{x + |x|(1 + x)}{x} \sin \left( \frac{1}{x} \right) \), for \( x \neq 0 \). Write \( L = \lim_{x \to 0^-} f(x) \) and \( R = \lim_{x \to 0^+} f(x) \). Then which one of the following is TRUE?
Show Hint
Check the behavior of absolute value functions carefully for left and right limits when dealing with piecewise functions.
Updated On: Dec 15, 2025
- \( L \) exists but \( R \) does not exist
- \( L \) does not exist but \( R \) exists
- Both \( L \) and \( R \) exist
- Neither \( L \) nor \( R \) exists
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Analyzing the limits.
First, consider the left-hand limit \( L \) as \( x \to 0^- \). We have the term \( |x| \), which behaves differently for negative \( x \), leading to the non-existence of the limit. For the right-hand limit \( R \), we have the term \( \sin \left( \frac{1}{x} \right) \), which oscillates but remains bounded as \( x \to 0^+ \).
Step 2: Conclusion.
Since the left-hand limit \( L \) exists, but the right-hand limit \( R \) does not exist, the correct answer is (A).
First, consider the left-hand limit \( L \) as \( x \to 0^- \). We have the term \( |x| \), which behaves differently for negative \( x \), leading to the non-existence of the limit. For the right-hand limit \( R \), we have the term \( \sin \left( \frac{1}{x} \right) \), which oscillates but remains bounded as \( x \to 0^+ \).
Step 2: Conclusion.
Since the left-hand limit \( L \) exists, but the right-hand limit \( R \) does not exist, the correct answer is (A).
Was this answer helpful?
0
0
Top IIT JAM MA Mathematics Questions
- The sum of the infinite series \[ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} \pi^{2n+1}}{2^{2n+1} (2n)!} \] is equal to
- For which one of the following choices of \( N(x, y) \), is the equation \[ (e^x \sin y - 2y \sin x) \, dx + N(x, y) \, dy = 0 \] an exact differential equation?
- Let \( f, g : \mathbb{R} \to \mathbb{R} \) be two functions defined by \[ f(x) = \begin{cases} x |x| \sin \frac{1}{x} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \] and \[ g(x) = \begin{cases} x^2 \sin \frac{1}{x} + x \cos \frac{1}{x} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \] Then, which one of the following is TRUE?
- Let \( f, g : \mathbb{R} \to \mathbb{R} \) be two functions defined by \[ f(x) = \begin{cases} |x|^{1/8} \sin \tfrac{1}{|x|} \cos x & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \] and \[ g(x) = \begin{cases} e^x \cos \tfrac{1}{x} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases} \] Then, which one of the following is TRUE?
- Which one of the following is the general solution of the differential equation \[ \frac{d^2 y}{dx^2} - 8 \frac{dy}{dx} + 16y = 2e^{4x} ? \]
View More Questions
Top IIT JAM MA Calculus Questions
- The value of \[ \int_0^1 \left( \int_0^{\sqrt{y}} 3e^{x^3} \, dx \right) dy \] is equal to
- Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be defined by \[ f(x, y) = e^{y}(x^2 + y^2) \quad \text{for all } (x, y) \in \mathbb{R}^2. \] Then, which one of the following is TRUE?
- Let \( \Omega \) be the bounded region in \( \mathbb{R}^3 \) lying in the first octant \( (x \geq 0, y \geq 0, z \geq 0) \), and bounded by the surfaces \( z = x^2 + y^2 \), \( z = 4 \), \( x = 0 \) and \( y = 0 \). Then, the volume of \( \Omega \) is equal to:
- Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be defined by \[ f(x, y) = \begin{cases} \frac{x^2 + y^5}{x^2 + y^4} & \text{if } (x, y) \neq (0, 0), \\ 0 & \text{if } (x, y) = (0, 0). \end{cases} \] Then, which of the following is/are TRUE?
- Let \( u_1 = (1, 0, 0, -1) \), \( u_2 = (2, 0, 0, -1) \), \( u_3 = (0, 0, 1, -1) \), \( u_4 = (0, 0, 0, 1) \) be elements in the real vector space \( \mathbb{R}^4 \). Then, which of the following is/are TRUE?
View More Questions
Top IIT JAM MA Questions
- Which one of the following groups has elements of order 1, 2, 3, 4, 5 but does not have an element of order greater than or equal to 6 ?
- Consider the group G = {A ∈ M2 (ℝ): AAT = I2} with respect to matrix multiplication. Let
Z(G) = {A ∈ G : AB = BA, for all B ∈ G}.
Then, the cardinality of Z(G) is - Let V be a nonzero subspace of the complex vector space 𝑀7(ℂ) such that every nonzero matrix in 𝑉 is invertible. Then, the dimension of V over ℂ is
- For \(n\ \in \N\), let
\(a_n=\frac{1}{(3n+2)(3n+4)}\) and \(b_n=\frac{n^3+\cos(3^n)}{3n+n^3}\).
Then, which one of the following is TRUE ? - Let \(a=\begin{bmatrix} \frac{1}{\sqrt3} \\ \frac{-1}{\sqrt2} \\ \frac{1}{\sqrt6} \\ 0\end{bmatrix}\). Consider the following two statements.
P : The matrix I4 - aaT is invertible.
Q: The matrix I4 - 2aaT is invertible.
Then, which one of the following holds ?
View More Questions