Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\cos2\text{x}-1}{\cos\text{x}-1}$

Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\cos2\text{x}-1}{\cos\text{x}-1}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{1-\cos2\text{x}}{1-\cos\text{x}}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{2\sin^2\text{x}}{2\sin^2\frac{\text{x}}{2}}$
$=\frac{\lim\limits_{\text{x} \rightarrow0}(\sin\text{x})^2}{\lim\limits_{\text{x} \rightarrow0}\big(\sin\frac{\text{x}}{2}\big)^2}$
$=\frac{\lim\limits_{\text{x} \rightarrow0}\big(2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}\big)^2}{\lim\limits_{\text{x} \rightarrow0}\big(\sin\frac{\text{x}}{2}\big)^2}$
$=4\lim\limits_{\text{x} \rightarrow0}\cos^2\frac{\text{x}}{2}$
$=4\times1$
$=4$

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

Similar questions

Express the following complex numbers in the form $\text{r}(\cos\theta+\text{i}\sin\theta):$
$\frac{1-\text{i}}{\cos\frac{\pi}{3}+\text{i}\sin\frac{\pi}{3}}$
If $\cos\theta+\tan\theta=2\text{cosec}\theta,$ then find the general value of $\theta.$
Match each item given under the column $C_1$ to its correct answer given under the column $C_2.$
There are $10$ professors and $20$ lecturers out of whom a committee of $2$ professors and $3$ lecturer is to be formed. Find:
 
$C_1$
  $C_2$
$(a)$
In how many ways committee can be formed.
$(i)$ $^{10}C_2 \times ^{19}C_3$
$(b)$
In how many ways a particular professor is included.
$(ii)$ $^{10}C_2 \times ^{19}C_2$
$(c)$
In how many ways a particular lecturer is included.
$(iii)$ $^9C_1 \times ^{20}C_3$
$(d)$
In how many ways a particular lecturer is excluded.
$(iv)$ $^{10}C_2\times ^{20}C_3$
$\sin5\text{x}=5\cos^4\text{x}\sin\text{x}-10\cos^2\text{x}\sin^3\text{x}+\sin^5\text{x}$
Find the equation of the circle concentric with the circle $x^2 + y^2 - 6x + 12y + 15 = 0$ and double of its area.
calculate the mean deviation from the mean for the following data:
13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17
Evaluate the following limits.
$\lim\limits_{\text{x} \rightarrow\pi}\frac{1-\sin\frac{\text{x}}{2}}{\cos\frac{\text{x}}{2}\Big(\cos\frac{\text{x}}{4}-\sin\frac{\text{x}}{4}\Big)}$ 
If $\sin\alpha+\sin\beta=\text{a}$ and $\cos\alpha+\cos\beta=\text{b},$ show that
$\sin(\alpha+\beta)=\frac{2\text{ab}}{\text{a}^2+\text{b}^2}$
A line is such that its segment between the straight lines 5x - y - 4 = 0 and 3x + 4y - 4 = 0 is bisected at the point (1, 5). Obtain its equation.
Mean and standard deviation of 100 observations were found to be 40 and 10, respectively. If at the time of calculation two observations were wrongly taken as 30 and 70 in place of 3 and 27 respectively, find the correct standard deviation.