Question:
The value of
\[ \lim_{x \to \infty} \frac{x^3 \sin\left(\frac{1}{x}\right) - 2x^2}{1 + 3x^2} \]
is ______.
The value of
\[ \lim_{x \to \infty} \frac{x^3 \sin\left(\frac{1}{x}\right) - 2x^2}{1 + 3x^2} \]
is ______.
Show Hint
For limits involving \( \sin(1/x) \), use small-angle approximation.
Updated On: Apr 30, 2026
- \( 0 \)
- \( \frac{1}{3} \)
- \( -1 \)
- \( -\frac{2}{3} \)
- \( -\frac{1}{3} \)
Show Solution
Verified By Collegedunia
The Correct Option is
Solution and Explanation
Concept:
Use:
\[
\sin x \approx x \text{for small } x
\]
Step 1: Approximate sine. \[ \sin\left(\frac{1}{x}\right) \approx \frac{1}{x} \] \[ x^3 \sin\left(\frac{1}{x}\right) \approx x^2 \]
Step 2: Simplify expression. \[ \frac{x^2 - 2x^2}{1 + 3x^2} = \frac{-x^2}{1+3x^2} \]
Step 3: Divide by \(x^2\). \[ = \frac{-1}{\frac{1}{x^2} + 3} \] \[ \to \frac{-1}{3} \]
Step 1: Approximate sine. \[ \sin\left(\frac{1}{x}\right) \approx \frac{1}{x} \] \[ x^3 \sin\left(\frac{1}{x}\right) \approx x^2 \]
Step 2: Simplify expression. \[ \frac{x^2 - 2x^2}{1 + 3x^2} = \frac{-x^2}{1+3x^2} \]
Step 3: Divide by \(x^2\). \[ = \frac{-1}{\frac{1}{x^2} + 3} \] \[ \to \frac{-1}{3} \]
Was this answer helpful?
0
0
Top KEAM Mathematics Questions
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
- The value of $ \cos [{{\tan }^{-1}}\{\sin ({{\cot }^{-1}}x)\}] $ is
- The solutions set of inequation $\cos^{-1}x < \,\sin^{-1}x$ is
- Let $\Delta= \begin{vmatrix}1&1&1\\ 1&-1-w^{2}&w^{2}\\ 1&w&w^{4}\end{vmatrix}$, where $w \neq 1$ is a complex number such that $w^3 = 1$. Then $\Delta$ equals
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
View More Questions
Top KEAM limits and derivatives Questions
- If $ f(x)=(x-2)(x-4)(x-6)....(x-2n), $ then $ f'(2) $ is
- \(\lim\limits_{t\rightarrow0}\frac{x}{\sqrt{9-x}-3}\) is equal to
- \(\lim\limits_{t\rightarrow-3}\frac{x^2+16x+39}{2x^2+7x+3}\) is equal to
- If h(x)=4x3-5x+7 is the derivative of f(x), then \(\lim\limits_{t\rightarrow0}\frac{f(1+t)-f(1)}{t}\) is equal to
- \(\lim\limits_{t\rightarrow0}\frac{tan^2(\frac{\pi}{3}+t)-3}{t}\) is equal to
View More Questions
Top KEAM Questions
- i.$\quad$ They help in respiration ii.$\quad$ They help in cell wall formation iii.$\quad$ They help in DNA replication iv. $\quad$They increase surface area of plasma membrane Which of the following prokaryotic structures has all the above roles?
- A body oscillates with SHM according to the equation (in SI units), $x = 5 cos \left(2\pi t +\frac{\pi}{4}\right) .$ Its instantaneous displacement at $t = 1$ second is
- The pH of a solution obtained by mixing 60 mL of 0.1 M BaOH solution at 40m of 0.15m HCI solution is
Kepler's second law (law of areas) of planetary motion leads to law of conservation of
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
View More Questions