Download App

Limits — MATHS STD 11 Science — Question

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

Answer

$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{2}}}\frac{1-\sin\text{x}}{\big(\frac{\pi}{2}-\text{x}\big)}$ If $\text{x}\rightarrow\frac{\pi}{2},\frac{\pi}{2}-\text{x}\rightarrow0$ Let $\frac\pi2-\text{x}=\text{y}$ they y → 0 $=\lim\limits_{\text{y}\rightarrow{0}}\frac{1-\sin\big(\frac{\pi}{4}-\text{y}\big)}{\text{y}^2}$ $=\lim\limits_{\text{y}\rightarrow{0}}\frac{1-\cos\text{y}}{\text{y}^2}$ $=\lim\limits_{\text{y}\rightarrow{0}}\frac{2\sin^2\frac{\text{y}}{2}}{\text{y}^2}$ $=2\Bigg(\lim\limits_{\text{y}\rightarrow{0}}\frac{\sin\frac{\text{y}}{2}}{\frac{\text{y}}{2}}\Bigg)^2\times\frac14$ $\Big[\because\lim\limits_{\text{x}\rightarrow{0}}\frac{\sin\text{x}}{\text{x}}=1\Big]$ $=2\times1\times\frac14$ $=\frac12$

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

A coin is tossed three times and the outcomes are recorded. How many possible outcomes are there? How many possible outcomes if the coin is tossed four times? Five times? n times?
→
Evaluate $\lim\limits_{\text{x}\rightarrow2}{\text{f(x)}}$ (if it exist), where $\text{f(x)}=\begin{cases}\text{x}-[\text{x}],&\text{x}<2\\4 ,& \text{x} = 2\\3\text{x}-5, & \text{x} > 2\end{cases}.$
→
If $\frac{\text{(2n!)}}{3!(2\text{n}-3)!}$ and $\frac{\text{n!}}{2!(2\text{n}-2)!}$ are in the ratio 44 : 3 find n.
→
Let A = {1, 2, 4, 5}, B = {2, 3, 5, 6}, C = {4, 5, 6, 7}. Verify the following identities: $\text{A}\cap(\text{B}\cup\text{C})=(\text{A}\cap\text{B})\cup(\text{A}\cap\text{C})$
→
Show that $\lim\limits_{\text{x}\rightarrow2^-}\ \frac{\text{x}}{[\text{x}]}\ne\lim\limits_{\text{x}\rightarrow2^+}\frac{\text{x}}{[\text{a}]}.$
→
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.
→
Solve the following system of equations in R. $|3-4\text{x}|\geq9$
→
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{1+\text{x}+\text{x}^2}-1}{\text{x}}$
→
Let A = {3, 6, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5 , 10, 15, 20}. Find:
  1. A - B
  2. A - C
  3. A - D
  4. B - A
  5. C - A
  6. D - A
  7. B - C
  8. B - D. 
→
Find the equation of the perpendicular bisector of the line joining the points (1, 3) and (3, 1).
→