Evaluate the following limit: _ x 0 2 - 2x x^3 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\text{x}-\sin2\text{x}}{\text{x}^3}$

✓

Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\text{x}-\sin2\text{x}}{\text{x}^3}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\text{x}-2\sin\text{x}\cos\text{x}}{\text{x}^3}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\text{x}(1-\cos\text{x})}{\text{x}^3}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\text{x}(1-\cos\text{x})}{\text{x}^3}\times\frac{1+\cos\text{x}}{1+\cos\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\big(1-\cos^2\text{x}\big)}{\text{x}^3(1+\cos\text{x})}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\big(\sin^2\text{x}\big)}{\text{x}^3(1+\cos\text{x})}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\sin^3\text{x}}{\text{x}^3(1+\cos\text{x})}$
$=2\lim\limits_{\text{x}\rightarrow0}\Big(\frac{\sin\text{x}}{\text{x}}\Big)^3\times\lim\limits_{\text{x}\rightarrow0}\frac{1}{(1+\cos\text{x})}$
$=2\times1\times\frac{1}{(1+1)}$
$=1$

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 "Solve the Following Question.(4 Marks)" from Limits→Limits — all question types→Maths STD 11 Science question papers→Maharashtra Board STD 11 Science question papers→Maharashtra Board question paper generator→

Similar questions

If ${^\text{28}}\text{C}_{2\text{r}}:{^\text{24}}\text{C}_{\text{2r-4}},=225:11,$ Find r.

Evaluate the following limit:
If $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{9}-\text{a}^9}{\text{x}-\text{a}}=\lim\limits_{\text{x}\rightarrow5}(4+\text{x}),$ find all possible value of a.

Find the equation of the straight line passing through the point of intersection of $2x + y - 1 = 0$ and $x + 3y - 2 = 0$ and making with the coordinate axes a triangle of area $\frac{3}{8}$ sq.units.

A natural number is chosen at random from amongst first 500. What is the probability that the number so chosen is divisible by 3 or 5?

Find the sum of the following geometric series:$(\text{x}+\text{y})+(\text{x}^2+\text{xy}+\text{y}^2)+(\text{x}^3+\text{x}^2\text{y}+\text{xy}^2+\text{y})+\ ...\text{ to n terms;}$

Calculate the mean deviation about the median of the following observation:
$34, 66, 30, 38, 44, 50, 40, 60, 42, 51$

Find the trigonometric functions of $: – 225^\circ$

Find the equation of the ellipse in standard form if : the distance between its directrices is 10 and which passes through (-√5, 2).

There are 10 professors and 20 students out of whom a committee of 2 professors and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees:

A particular professor is included.
A particular student is included.
A particular student is excluded.

Find the locus of $P$ if $PA^2 + PB^2 = 2k^2,$ where A and B are the points $(3, 4, 5)$ and $(-1, 3, -7).$