Evaluate the following limit: _ 0 1- 4 1- 6 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\theta\rightarrow0}\frac{1-\cos4\theta}{1-\cos6\theta}$

✓

Answer

$\lim\limits_{\theta\rightarrow0}\frac{1-\cos4\theta}{1-\cos6\theta}$
$=\lim\limits_{\theta \rightarrow0}\frac{2\sin^22\theta}{2\sin^23\theta}$
$=\lim\limits_{\theta \rightarrow0}\frac{(\sin2\theta)^2}{(\sin3\theta)^2}$
$=\frac{\lim\limits_{\theta \rightarrow0}\big(\frac{\sin2\theta}{2\theta}\big)^2\times4\theta^2}{\lim\limits_{\theta \rightarrow0}\big(\frac{\sin3\theta}{3\theta}\big)^2\times9\theta^2}$
$=\frac{1\times4\theta^2}{1\times9\theta^2}$
$=\frac{4}{9}$

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 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

Solve the inequality: $-12<4-\frac{3 x}{-5} \leq 2$

Does the point $(-2.5, 3.5)$ lie inside, outside or on the circle $x^2 + y^2 = 25?$

If a, b, c are in G.P., Prove that $\log \text{a}, \log \text{b},\log\text{c}$ are in A.P.

If $A=\{1,2,3), B=\{3,4\}$ and $C=\{4,5,6\}$, find
i. $A \times(B \cap C)$
ii. $(A \times B) \cap(A \times C)$

State the following statement is true and false. Justify your answer.
496 $\notin$ {y | the sum of all the positive factors of y is 2y}.

Find the equation of the parabola that satisfies the given conditions: Vertex $(0, 0)$ Focus $(3, 0)$

Solve the inequality 4x + 3 < 5x + 7 for real x.

Solve the inequality $x+\frac{x}{2}+\frac{x}{3}<11$ for real x.

Prove that: $\tan\frac{\pi}{12}+\tan\frac{\pi}{6}+\tan\frac{\pi}{12}\tan\frac{\pi}{6}=1$

Which term in the expansion of $\Bigg\{\Big(\frac{\text{x}}{\sqrt{\text{y}}}\Big)^\frac{1}{3}+\bigg(\frac{\text{y}}{\text{x}^\frac{1}{3}}\bigg)^\frac{1}{2}\Bigg\}^{21}$ contains x and y to one and the same power?