Evaluate the following limit: _ x 0 x + x^2+ - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}\cos\text{x}+\sin\text{x}}{\text{x}^2+\tan\text{x}}$

✓

Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}\cos\text{x}+\sin\text{x}}{\text{x}^2+\tan\text{x}}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{\cos\text{x}+\frac{\sin\text{x}}{\text{x}}}{\text{x}+\frac{\tan\text{x}}{\text{x}}}$
$=\frac{\lim\limits_{\text{x} \rightarrow0}\cos\text{x}+\lim\limits_{\text{x} \rightarrow0}\frac{\sin\text{x}}{\text{x}}}{\lim\limits_{\text{x} \rightarrow0}\text{x}+\lim\limits_{\text{x} \rightarrow0}\frac{\tan\text{x}}{\text{x}}}$
$=\frac{1+1}{0+1}=\frac21$
$=2$

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

Define a relation $R$ on the set $N$ of natural number by $R = \{(x, y): y = x + 5\}, x$ is a natural number less than $4, \text{x, y}\in\text{N}\}$
Depict this relationship using:

Roster form.
An arrow diagram. Write down the domain and range or $R.$

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

For each of the following compound statements first identify the connecting words and then break it into component statements.
The sand heats up quickly in the Sun and does not cool down fast at night.

Find the indicated terms of the sequence, whose nth term is $a_n = 4n - 3; a_{17}, a_{24}.$

Find the equation of the perpendicular bisector of the line joining the points $(1, 3)$ and $(3, 1).$

The runs scored by players in $10$ matches are $38, 70, 48, 34, 42, 55, 63, 46, 54, 44.$ Find the variance and standard deviation.

Find the multiplicative inverse of the complex numbers $ = \sqrt 5 + 3i$

Find the domain of the following real valued functions of real variable:
$\text{f(x)}=\frac{\text{x}^2+2\text{x}+1}{\text{x}^2-8\text{x}+12}$

If in two circles, arcs of the same length subtend angles $60^\circ$ and $75^\circ$ at the centre, find the ratio of their radii.

The numbers $1, 2, 3$ and $4$ are written separately on four slips of paper. The slips are then put in a box and mixed thoroughly. A person draws two slips from the box, one after the other, without replacement.
Describe the following events:
$A =$ The number on the first slip is larger than the one on the second slip.
$B =$ The number on the second slip is greater than $2.$
$C =$ The sum of the numbers on the two slips is $6$ or $7$.
$D =$ The number on the second slips is twice that on the first slip.
Which pair$(s)$ of events is $($are$)$ mutually exclusive?