Evaluate the following limit: _ x 2 2- 1+ ^2x - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{2}}}\frac{\sqrt{2}-\sqrt{1+\sin\text{x}}}{\cos^2\text{x}}$

✓

Answer

$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{2}}}\frac{\sqrt{2}-\sqrt{1+\sin\text{x}}}{\cos^2\text{x}}$$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{2}}}\frac{\sqrt{2}-\sqrt{1+\sin\text{x}}}{\cos^2\text{x}}\frac{\sqrt{2}+\sqrt{1+\sin\text{x}}}{\sqrt{2}+\sqrt{1+\sin\text{x}}}$
$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{2}}}\frac{2-1-\sin\text{x}}{\cos^2\text{x}\big(\sqrt{2}-\sqrt{1+\sin\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{2}}}\frac{1-\sin\text{x}}{\big(1-\sin^2\text{x}\big)\big(\sqrt{2}-\sqrt{1+\sin\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{2}}}\frac{1}{\big(1+\sin\text{x}\big)\big(\sqrt{2}+\sqrt{1+\sin\text{x}}\big)}$
$=\frac{1}{(1+1)\big(\sqrt{2}+\sqrt{2}\big)}$
$=\frac{1}{\big(4\sqrt{2}\big)}$

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 "3 Marks Question" from Limits→Limits — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Find the distance between the following pairs of points:
$(i) (2, 3, 5)$ and $(4, 3, 1)$
$(ii) (-3, 7, 2)$ and $(2, 4, -1)$
$(iii) (-1, 3, -4)$ and $(1, -3, 4)$
$(iv) (2, -1, 3)$ and $(-2, 1, 3)$

Find the conjugates of the following complex numbers:
$\frac{(1+\text{i})(2+\text{i})}{3+\text{i}}$

Find the middle terms(s) in the expansion of:
$\Big(\frac{\text{x}}{3}+9\text{y}\Big)^{10}$

Prove that the line y -x + 2 = 0 divides the join of points (3, -1) and (8, 9) in the ratio 2 : 3.

A man wants to cut three lengths from a single piece of board of length 91cm. The second length is to be 3cm longer than the shortest and the third length is to be twice as long as the shortest. What are the possible lengths of the shortest board if the third piece is to be at least 5cm longer than the second?
[Hint: If x is the length of the shortest board, then x , (x + 3) and 2x are the lengths of the second and third piece, respectively. Thus, x + (x + 3) + 2x $\leq$ 91 and 2x $\ge$ (x + 3) + 5].

If the points A(3, 2, -4), B(9, 8, -10) and C(5, 4, -6) are collinear, find the ratio in which C divides AB.

Verify that $(0, 7, 10), (-1, 6, 6)$ and $(-4, 9, 6)$ are the vertices of a right$-$angled triangle.

Reduce the following equations to the normal form and find p and $\alpha$ in each case:
$\text{x}-\text{y}+2\sqrt{2}=0$

For any two sets, prove that:
$\text{A}\cup(\text{A}\cap\text{B})=\text{A}.$

Let A = {x : x $\in$ N}, B = {x : x = 2n, n $\in$ N}, C = {x : x = 2n - 1, n $\in$ N} and D = {x : x is a prime natural number}. Find:
$\text{B}\cap\text{D}$