Evaluate _ x 2 1+ 2 x ( -2 x)^2 . - Vidyadip

Question

Evaluate $\lim _{x \rightarrow \frac{\pi}{2}} \frac{1+\cos 2 x}{(\pi-2 x)^2}.$

✓

Answer

$\lim _{x \rightarrow \frac{\pi}{2}} \frac{1+\cos 2 x}{(\pi-2 x)^2}$
Let $\frac{\pi}{2}+h=x$
$\begin{array}{l}\because \text { If } x=\frac{\pi}{2}, \text { then } h \rightarrow 0 \\=\lim _{h \rightarrow 0} \frac{1+\cos \left\{2\left(\frac{1}{2} \pi+h\right)\right\}}{\left\{\pi-2\left(\frac{1}{2}\pi+h\right)\right\}^2}\left[\because \lim _{x \rightarrow a} f(x)=\lim _{h \rightarrow 0} f(a+h)\right] \\=\lim _{h \rightarrow 0} \frac{1+\cos (\pi+2 h)}{4 h^2}=\lim _{h \rightarrow 0} \frac{1-\cos 2 h}{4 h^2}\end{array}$
$\begin{array}{l}=\lim _{h \rightarrow 0} \frac{2 \sin ^2 h}{4 h^2}=\frac{1}{2} \lim _{h \rightarrow 0}\left(\frac{\sin h}{h}\right)^2 \\ =\frac{1}{2} \cdot(1)^2=\frac{1}{2}\end{array}$

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 "2 Marks Questions" from PART - 2 CH - 12 Limits and Derivatives→PART - 2 CH - 12 Limits and Derivatives — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

A five digit number is formed by the digits 1, 2, 3, 4, 5 without repetition. Find the probability that the number is divisible by 4.

The sum of three numbers which are consecutive terms of an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.

Find the equation of the circle whose centre lies on the positive direction of y-axis at a distance 6 from the origin and whose radius is 4.

Which term of the sequence $24,\ 23\frac{1}{4},\ 22\frac{1}{2},\ 21\frac{3}{4}...$ is the first negative term?

Find the slope of a line, which passes through the origin and mid-point of the line segment joining the points P(0, - 4) and B(8, 0).

Prove that:
$(\sin^3\text{x}+\sin\text{x})\sin\text{x}+(\cos3\text{x}-\cos\text{x})\cos\text{x}=0$

Find $\mathop {\lim }\limits_{x \to 1} f(x)$ where f(x) = $\left\{ {\begin{array}{*{20}{c}} {{x^2} - 1,}&{x \le 0} \\ { - {x^2} - 1,}&{x > 1} \end{array}} \right.$

Find the equation of hyperbola having Foci ($\pm$4, 0) and the latus rectum is of length 12.

Prove $\left( A ^{\prime}\right)^{\prime}= A$ for an empty set $A$.

Without using distance formula, show that the points (-2, -1), (4, 0), (3, 3) and (-3, 2) are the vertices of a parallelogram.