L'Hospital Rule - Previous Year Questions
Previous Year Questions on L'Hospital Rule
L'Hospital Rule has been a cornerstone in solving indeterminate forms in calculus. Its history dates back to the contributions of Guillaume de l'Hôpital, who first formalized the rule in his book "Analyse des Infiniment Petits". The importance of L'Hospital Rule lies in its ability to simplify complex limits, which are extensively applied in mathematical modeling and real-world problem-solving.
Vidyasagar University
2023-24 (NEP)
- Verify that \(\lim\limits_{x\to \infty}\left[ x-x^{2}\log\left(1+\frac{1}{x} \right) \right]=\frac{1}{2} \). [2]
- Your answer find the value of \(\lim\limits_{x\to 1}x^{\frac{1}{x-1}} \). [2]
- Find the value of \(a\) and \(b\) so that \(\lim\limits_{x\to 0}\frac{a\sin{2x}-b\sin{x}}{x^{3}}=1 \). [5]
- Find \(a,b,c\) such that \(\frac{ae^{x}-b\cos{x}+ce^{-x}}{x\sin{x}}\to 2 \) as \(x\to 0 \). [4]
- Find the value of \(\lim\limits_{x\to \infty}\left[a_{0}x^{m}+a_{1}x^{m-1}+…+a_{m} \right]^{\frac{1}{x}} \), \(m\) being a positive integer and \(a_{0}\ne 0 \). [2]
- Evaluate the following limits \(\lim\limits_{x\to n}x\log\left(\sin{x} \right) \) in \(\left(0,\pi \right)\). [4]
- No Questions
- If \(n\) be any positive integer, find the value of \(\lim\limits_{x\to n}x^{\frac{x-n}{\sin{\pi x}}} \). [2]
- Find the value of \(a\) and \(b\) so that \(\lim\limits_{x\to 0}\frac{a\sin{2x}-b\sin{x}}{x^{3}}=1 \). [3]
- Your answer find the value of \(\lim\limits_{x\to 1}x^{\frac{1}{x-1}} \). [2]
- If \(\alpha,\beta \) be the roots of the equation \(ax^{2}++bx+c=0 \) then show that \(\lim\limits_{x\to \alpha}x^{\frac{1-\cos{\left( ax^{2}++bx+c\right)}}{\left(x-\alpha\right)^{2}}}=\frac{1}{2}a^{2}\left(\alpha-\beta\right)^{2} \). [5]
FAQs
- What is L’Hospital Rule?
It is a mathematical rule used to evaluate indeterminate forms like 0/0 or ∞/∞ by differentiating the numerator and denominator. - When was L’Hospital Rule introduced?
It was first formalized in 1696 in Guillaume de l’Hôpital’s book. - How is L’Hospital Rule applied in real life?
It is used in engineering, physics, and economics to solve complex limit problems. - Is L’Hospital Rule part of university curricula?
Yes, it is included in various university courses on calculus. - Can L’Hospital Rule be applied to all indeterminate forms?
No, it is specifically for 0/0 and ∞/∞ forms. - What are some limitations of L’Hospital Rule?
It requires differentiable functions and may not work if limits do not exist. - What topics are related to L’Hospital Rule?
Topics include Linear Algebra, Classical Algebra, and Abstract Algebra. - Can L’Hospital Rule solve improper integrals?
Yes, it can be extended to certain cases involving improper integrals. - Is L’Hospital Rule applicable in higher mathematics?
Yes, it is applied in advanced topics like analysis and mathematical modeling. - Where can I find practice questions for L’Hospital Rule?
You can explore Mathematics Questions or specific topics like Differential Calculus.
Related Questions
Trending Today
Semeter-1 Mathematics Honours (Vidyasagar University)
[manual_filtered_papers subject="Math" university="VU" semester="1"]
Previous Year's Mathematics Honours (Vidyasagar University) Questions papers
[manual_filtered_papers subject="Math" university="VU"]
Categories
[categories]
Related Notes
Related Questions
Related Quizzes
No posts found!
