Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\cos3\text{x}-\cos5\text{x}}{\text{x}^2}$

Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\cos3\text{x}-\cos5\text{x}}{\text{x}^2}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\Big(-2\sin\big(\frac{3\text{x}+5\text{x}}{2}\big)\sin\big(\frac{3\text{x}-5\text{x}}{2}\big)\Big)}{\text{x}^2}$
$=\lim\limits_{\text{x}\rightarrow0}\Big(\frac{-2\sin4\text{x}\sin(-\text{x})}{\text{x}^2}\Big)$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\sin4\text{x}\sin\text{x}}{\text{x}^2}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\sin4\text{x}}{\text{x}}\times\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}$
$=2\Big(\lim\limits_{\text{x}\rightarrow0}\frac{\sin4\text{x}}{4\text{x}}\times4\Big)\times\Big(\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}\Big)$ $\Big[\because\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}=1\Big]$
$=8$

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

In each the following find the equation of the hyperbola satisfying the given conditions:
Foci $(0, \pm13), $ conjugate axis = 24
Sum the following series to n terms:
$1 + 4 + 13 + 40 + 121 + .....$
Find the general solution for each of the following equations:
$\sin\text{x}+\sin3\text{x}+\sin5\text{x}=0$
Match the following sets for all sets $A, B$ and $C.$
$(i)$ $((\text{A}'\cup\text{B}')-\text{A})'$ $(a)$ $\text{A} - \text{B}$
$(ii)$ $[\text{B}'\cup(\text{B}'-\text{A})]'$ $(b)$ $\text{A}$
$(iii)$ $(\text{A} - \text{B}) - (\text{B} - \text{C})$ $(c)$ $\text{B}$
$(iv)$ $(\text{A}-\text{B})\cap(\text{C}-\text{B})$ $(d)$ $(\text{A}\times\text{B})\cap(\text{A}\times\text{C})$
$(v)$ $\text{A}\times(\text{B}\cap\text{C})$ $(e)$ $(\text{A}\times\text{B})\cup(\text{A}\times\text{C})$
$(vi)$ $\text{A}\times(\text{B}\cup\text{C})$ $(f)$ $(\text{A}\cap\text{C})-\text{B}$
Prove that the number of subsets of a set containing n distinct elements is $2^n $for all $\text{n}\in\text{N}.$
Prove the following identities:
$\frac{\cos\text{x}}{1-\sin\text{x}}=\frac{1+\cos\text{x}+\sin\text{x}}{1+\cos\text{x}-\sin\text{x}}$
Find the coefficient of variation for the following data:
Size (in cms):
10-15
15-20
20-25
25-30 30-35 35-40
No. of items:
2
 
8
20
35 20 15
Find the equation to the ellipse in the following case:Vertices $(\pm6, 0),$ foci $(\pm4, 0)$
Find the equation of the line passing through the point (-3, 5) and perpendicular to the line joining (2, 5) and (-3, 6).
Evaluate the following limit:
Evaluate: $\lim\limits_{\text{n}\rightarrow\infty}\frac{1^4+2^4+3^4+\ \cdots+\text{n}^4}{\text{n}^5}-\lim\limits_{\text{n}\rightarrow\infty}\frac{1^3+2^3+\ \cdots+\text{n}^3}{\text{n}^5}$