Lim x - 0 cos 3 x-cos 5 x x 2

Limx #8594;0cos #160;3x #160; cos #160;5xx2 = #160;limx #8594;0 2sin3x+5x2 #160;sin3x 5x2x2 #160; #160; #160; #160; #160; #160; #1 ...

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Byju's Answer

Standard XII

Mathematics

Graph of Trigonometric Ratios

lim x → 0 cos...

Question

$\lim_{x \to 0} \frac{\cos 3 x - \cos 5 x}{x^{2}}$

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Solution

$\lim_{x \to 0} [\frac{\cos 3 x - \cos 5 x}{x^{2}}]$

$= \lim_{x \to 0} [\frac{- 2 \sin (\frac{3 x + 5 x}{2}) \sin (\frac{3 x - 5 x}{2})}{x^{2}}] [∵ cosC - cosD = - 2 \sin (\frac{C + D}{2}) \sin (\frac{C - D}{2})] = 2 \lim_{x \to 0} [\frac{2 \sin 4 x \sin x}{x^{2}}] [∵ \sin (- θ) = - \sin θ] = 2 \lim_{x \to 0} [\frac{\sin 4 x}{4 x} \times 4 \times \frac{\sin x}{x}] = 2 \times 4 = 8 .$

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Similar questions

lim x \to 0 \frac{cos 3 x - cos 7 x}{x^{2}}

Q. solve

{lim}_{x \to 0} \frac{cos 3 x - cos x}{x^{2}}

Q. STATEMENT-1 :

{lim}_{x \to \infty} (\frac{1^{2}}{x^{3}} + \frac{2^{2}}{x^{3}} + \frac{3^{2}}{x^{3}} + . . . . . . . + \frac{x^{2}}{x^{3}}) = lim x \to \infty \frac{1^{2}}{x^{3}} + lim x \to \infty \frac{2^{2}}{x^{3}} + . . . . . + lim x \to \infty \frac{x^{2}}{x^{3}} = 0

STATEMENT-2 :

{lim}_{x \to a} (f_{1} (x) + f_{2} (x) + . . . . . + f_{n} (x)) = {lim}_{x \to a} f_{1} (x) + {lim}_{x \to a} f_{2} (x) + . . . . + {lim}_{x \to a} f_{n} (x)

, where

n \in N .

If both $lim x \to a f (x)$ and $lim x \to a g (x)$ and exist finitely and $lim x \to a g (x) = 0$ , then
$lim x \to a \frac{f (x)}{g (x)} = \frac{lim x \to a f (x)}{lim x \to a g (x)}$

$lim x \to a [f (x) + g (x)] = lim x \to a f (x) + lim x \to a g (x)$
is valid if $lim x \to a f (x)$ does not exists?

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limx→0cos 3x -cos 5xx2

limx→0cos 3x -cos 5xx2 = limx→0-2sin3x+5x2 sin3x-5x2x2 ∵cosC-cosD=-2sinC+D2sinC-D2= 2 limx→02sin 4x sin xx2 ∵sin-θ=-sinθ= 2limx→0sin 4x4x×4×sinxx= 2×4= 8.

$\lim_{x \to 0} \frac{\cos 3 x - \cos 5 x}{x^{2}}$