GATE CE 2014 Set 1 — Question 11
Go beyond PYQs with Success TrackerAI-powered personalised practice and doubt support. Unlimited practice on eligible plans; AI usage limits apply.MCQ+1 / -0.33EasyLimits & ContinuityCalculusEngineering Mathematics
Engineering Mathematics → Calculus → Limits & Continuity
Last updated
Question
equals to
Correct answer
(C) 1
Solution
To evaluate the limit L = \lim_{x \to $\infty$} \left( \frac{x + \sin x}{x} \right)we can simplify the expression inside the limit by dividing each term in the numerator by the denominator:$\frac{x + \sin x}{x} = \frac{x}{x} + \frac{\sin x}{x} = 1 + \frac{\sin x}{x}$Now, we apply the limit to each term:L = \lim_{x \to $\infty$} \left( 1 + \frac{\sin x}{x} \right) = \lim_{x \to $\infty$} (1) + \lim_{x \to $\infty$} \left( \frac{\sin x}{x} \right)To evaluate the second limit, we use the Squeeze Theorem:
1.The sine function is bounded for all real :
2.For , dividing the entire inequality by gives:
$ -\frac{1}{x} \le \frac{\sin x}{x} \le \frac{1}{x}$3.Taking the limit as :
\lim_{x \to $\infty$} \left( -\frac{1}{x} \right) = 0 \quad \text{and} \quad \lim_{x \to $\infty$} \left( \frac{1}{x} \right) = 0By the Squeeze Theorem, it follows that:\lim_{x \to $\infty$} \left( \frac{\sin x}{x} \right) = 0Substituting this back into the original limit expression:Correct Option: CContinue learning with Success Tracker
A step still unclear? Work through it with support
Use Success Tracker to ask about the reasoning, then try another GATE CE question to check your understanding.
AI-powered practice· Unlimited practice on eligible plans
- PYQs with solutions
- Attempt available previous-year questions, then compare your reasoning with the worked solution. Coverage varies by stream.
- Practice that adapts
- Choose a topic, work on weaker areas and bookmark questions to revisit. Your attempts feed your progress tracking.
- AI doubt support
- Ask follow-up questions about a step or concept while practising, instead of stopping at the final answer.
Unlimited practice is available on eligible plans. Free practice and AI usage have limits; check the current plan allowances before choosing.
This page stays readable without an account. AI responses can be wrong; check them against the solution and source material.