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

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sin(2+\text{x})-\sin(2-\text{x})}{\text{x}}$

✓

Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sin(2+\text{x})-\sin(2-\text{x})}{\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}2\cos\frac{\Big(\frac{{2+\text{x}+2-\text{x}}{2}}{\text{x}}\Big)\times\sin\Big(\frac{2+\text{x}-2+\text{x}}{2}\Big)}{{\text{x}}}$
$=2\lim\limits_{\text{x}\rightarrow0}\frac{\cos(2)\times\sin\text{x}}{\text{x}}$
$=2\lim\limits_{\text{x}\rightarrow0}\cos2\times\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}$ $\Big[\because\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}=1\Big]$
$=2\cos2\times1$
$=2\cos2$

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.(3 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

Find the equation of a circle with a radius of 4 units and touch both the co-ordinate axes having centre in the third quadrant.

Line through the points (-2, 6) and (4, 8) is perpendicular to the line through the points (8, 12) and (x, 24). Find the value of x.

If A = {1, 2}, from the set A × A × A.

In how many ways can 5 different books be arranged on a shelf if (i) there are no restrictions

(II)2 books are always together

(III)2 books are never together

Express the following complex numbers in polar form and exponential form.

-1 + √3 i

Show that $x=-1$ is a tangent to circle $x^2+y^2-4 x-2 y-4=0$ at $(-1,1)$.

A coin is tossed three times and the outcomes are recorded. How many possible outcomes are there? How many possible outcomes if the coin is tossed four times? Five times? n times?

If the tangents drawn from the point $(-6,9)$ to the parabola $y^2=k x$ are perpendicular to

each other, find k.

Solve the following linear inequations in R:
$\frac{5\text{x}+8}{4-\text{x}}<2$

Two cards are drawn from a pack of 52 cards one after other without replacement. What is the probability that both cards are ace cards?