Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}\cos\text{x}+\sin\text{x}}{\text{x}^2+\tan\text{x}}$
$\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}\cos\text{x}+\sin\text{x}}{\text{x}^2+\tan\text{x}}$
✓
Answer
$\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}\cos\text{x}+\sin\text{x}}{\text{x}^2+\tan\text{x}}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{\cos\text{x}+\frac{\sin\text{x}}{\text{x}}}{\text{x}+\frac{\tan\text{x}}{\text{x}}}$
$=\frac{\lim\limits_{\text{x} \rightarrow0}\cos\text{x}+\lim\limits_{\text{x} \rightarrow0}\frac{\sin\text{x}}{\text{x}}}{\lim\limits_{\text{x} \rightarrow0}\text{x}+\lim\limits_{\text{x} \rightarrow0}\frac{\tan\text{x}}{\text{x}}}$
$=\frac{1+1}{0+1}=\frac21$
$=2$
$=\lim\limits_{\text{x} \rightarrow0}\frac{\cos\text{x}+\frac{\sin\text{x}}{\text{x}}}{\text{x}+\frac{\tan\text{x}}{\text{x}}}$
$=\frac{\lim\limits_{\text{x} \rightarrow0}\cos\text{x}+\lim\limits_{\text{x} \rightarrow0}\frac{\sin\text{x}}{\text{x}}}{\lim\limits_{\text{x} \rightarrow0}\text{x}+\lim\limits_{\text{x} \rightarrow0}\frac{\tan\text{x}}{\text{x}}}$
$=\frac{1+1}{0+1}=\frac21$
$=2$
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
Find the square root of the following complex numbers:
$8-15\text{i}$
→$8-15\text{i}$
The mean and standard deviations of the two brands of watches are given below:
Calculate the coefficient of variation of the two brands and interpret the results.
For the following G.P.s, find $S_n$.
→$p, q, \frac{ q ^2}{ p }, \frac{ q ^3}{ p ^2}, \ldots$
Write the first five terms in the following sequences:$\text{a}_1=1,\text{a}_\text{n}=\text{a}_{\text{n}-1}+2,\text{n}>1$
→Find the number of distinct words formed from letters in the word INDIAN. How many of them have the two N’s together?
Find the focal distance of a point on the parabola $\mathrm{y}^2=16 \mathrm{x}$ whose ordinate is 2 times the
→abscissa.
List all the element of the following sets:
E = {x : x is a month of a year not having 31 days};
E = {x : x is a month of a year not having 31 days};
Express the following in the form of $a + ib, a, b ∈ R, i = \sqrt{-1}.$ State the values of $a$ and $b:$
$\frac{2+\sqrt{-3}}{4+\sqrt{-3}}$
→$\frac{2+\sqrt{-3}}{4+\sqrt{-3}}$
Solve using intermediate value theorem:
Show that $5^x-6 x=0$ has a root in $[1,2]$
→Show that $5^x-6 x=0$ has a root in $[1,2]$
Prove the following : $\frac{\operatorname{cosec}\left(90^{\circ}-x\right) \sin \left(180^{\circ}-x\right) \cot \left(360^{\circ}-x\right)}{\sec \left(180^{\circ}+x\right) \tan \left(90^{\circ}+x\right) \sin (-x)}=1$
→