Download App

Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits3 Marks
Question
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}^2\big(1-\cos\text{x}^2\big)}{\text{x}^6}$
✓

Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}^2\big(1-\cos\text{x}^2\big)}{\text{x}^6}$ $=\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}^2\times2\sin^2\frac{\text{x}^2}{2}}{\text{x}^6}$ $=\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}^2}{\text{x}^2}\times\lim\limits_{\text{x}\rightarrow0}\frac{2\sin^2\frac{\text{x}^2}{2}}{\text{x}^4}$ $=\Big(\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}^2}{\text{x}}\Big)\times2\times\Bigg(\lim\limits_{\text{x}\rightarrow0}\frac{\sin\frac{\text{x}^2}{2}}{\frac{\text{x}^2}{2}}\Bigg)\times\frac14$ $=(1)^2\times2\times1\times\frac14$ $\Big[\because\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}=1\Big]$ $=\frac12$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "(Each question 3 marks)" from Limits→Limits — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the sum to n terms of the series $1 \times 2 \times 3 + 2 \times 3 \times 4 + 3 \times 4 \times 5 + \ldots$
→
Find the equation of the circle passing through the points: $(5, -8), (-2, 9)$ and $(2, 1)$
→
A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.
→
Find the equation of the lines through the point (3, 2) which make an angle of 45°with the line x - 2y = 3.
→
Solve the inequality and represent the solution graphically on number line: 3x – 7 > 2 (x – 6) , 6 – x > 11 – 2x
→
The sum of first three terms of a G.P. is $\frac{39}{10}$ and their product is 1. Find the common ratio and the terms.
→
In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?
→
The number of terms of an A.P. is even; the sum of odd terms is 24, of the even terms is 30, and the last term exceeds the first by $10\frac{1}{2},$ find the number of terms and the series.
→
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow{\infty}}{\sqrt{\text{x}+1}-\sqrt{\text{x}}}{}$
→
If 12th term of an A.P. is -13 and the sum of the first four terms is 24, what is the sum of first 10 terms?
→