Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\cos2\text{x}-1}{\cos\text{x}-1}$
$\lim\limits_{\text{x}\rightarrow0}\frac{\cos2\text{x}-1}{\cos\text{x}-1}$
✓
Answer
$\lim\limits_{\text{x}\rightarrow0}\frac{\cos2\text{x}-1}{\cos\text{x}-1}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{1-\cos2\text{x}}{1-\cos\text{x}}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{2\sin^2\text{x}}{2\sin^2\frac{\text{x}}{2}}$
$=\frac{\lim\limits_{\text{x} \rightarrow0}(\sin\text{x})^2}{\lim\limits_{\text{x} \rightarrow0}\big(\sin\frac{\text{x}}{2}\big)^2}$
$=\frac{\lim\limits_{\text{x} \rightarrow0}\big(2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}\big)^2}{\lim\limits_{\text{x} \rightarrow0}\big(\sin\frac{\text{x}}{2}\big)^2}$
$=4\lim\limits_{\text{x} \rightarrow0}\cos^2\frac{\text{x}}{2}$
$=4\times1$
$=4$
$=\lim\limits_{\text{x} \rightarrow0}\frac{1-\cos2\text{x}}{1-\cos\text{x}}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{2\sin^2\text{x}}{2\sin^2\frac{\text{x}}{2}}$
$=\frac{\lim\limits_{\text{x} \rightarrow0}(\sin\text{x})^2}{\lim\limits_{\text{x} \rightarrow0}\big(\sin\frac{\text{x}}{2}\big)^2}$
$=\frac{\lim\limits_{\text{x} \rightarrow0}\big(2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}\big)^2}{\lim\limits_{\text{x} \rightarrow0}\big(\sin\frac{\text{x}}{2}\big)^2}$
$=4\lim\limits_{\text{x} \rightarrow0}\cos^2\frac{\text{x}}{2}$
$=4\times1$
$=4$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Solve the following equations:
$\tan\text{x}+\tan2\text{x}+\tan3\text{x}=0$
→$\tan\text{x}+\tan2\text{x}+\tan3\text{x}=0$
Prove the following identities: $\frac{(1+\cot\text{x}+\tan\text{x})(\sin\text{x}+\cos\text{x})}{\sec^3\text{x}-\text{cosec}^3\text{ x}}=\sin^2\text{x}\cos^2\text{x}$
→Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\frac{2}{3}}-\text{a}^{\frac{2}{3}}}{\text{x}^{\frac{3}{4}}-\text{a}^{\frac{3}{4}}}$
→$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\frac{2}{3}}-\text{a}^{\frac{2}{3}}}{\text{x}^{\frac{3}{4}}-\text{a}^{\frac{3}{4}}}$
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow-1}\frac{\text{x}^2-\text{x}-2}{\big(\text{x}^2+\text{x}\big)+\sin(\text{x}+1)}$
→$\lim\limits_{\text{x}\rightarrow-1}\frac{\text{x}^2-\text{x}-2}{\big(\text{x}^2+\text{x}\big)+\sin(\text{x}+1)}$
Differentiate the following from first principle$\tan(\text{2x}+1)$
→A die is rolled 30 times and the following distribution is obtained. Find the variance and S.D.
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{2}-\sqrt{1+\cos\text{x}}}{\sin^2\text{x}}$
→$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{2}-\sqrt{1+\cos\text{x}}}{\sin^2\text{x}}$
Discuss whether the function f(x) = |x + 1| + |x – 1| is differentiable ∀ x ∈ R.
Tangents are drawn through a point $P$ to the ellipse $4 x^2+5 y^2=20$ having inclinations $\theta_1$and $\theta_2$ such that $\tan \theta_1+\tan \theta_2=2$. Find the equation of the locus of $P$.
→How many three-digit numbers can be formed from the digits 0, 1, 3, 5, 6 if repetitions of digits (i) are allowed (ii) are not allowed?