Limit of Fractional Sine

Calculus Level 1

lim ⁡ x → 0 sin ⁡ ( a x ) sin ⁡ ( b x ) = ? \displaystyle \lim_{x \to 0} \dfrac{ \sin(ax)}{\sin(bx)}=?

1 a/b 0 b/a not defined

Divyansh Chauhan by divyansh chauhan , 22, India

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2 solutions

Micah Wood
Nov 3, 2015

Using small-angle approximation , we have lim ⁡ x → 0 sin ⁡ ( a x ) sin ⁡ ( b x ) = lim ⁡ x → 0 a x b x = lim ⁡ x → 0 a b = a b \lim_{x\to 0}\dfrac{\sin(ax)}{\sin(bx)}=\lim_{x\to 0}\dfrac{ax}{bx} = \lim_{x\to 0}\dfrac ab = \boxed{\dfrac ab}

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Otto Bretscher
Nov 3, 2015

lim ⁡ x → 0 sin ⁡ ( a x ) sin ⁡ ( b x ) = lim ⁡ x → 0 ( sin ⁡ ( a x ) a x b x sin ⁡ ( b x ) a b ) = a b \lim_{x\to 0}\frac{\sin(ax)}{\sin(bx)}=\lim_{x\to 0}\left(\frac{\sin(ax)}{ax}\frac{bx}{\sin(bx)}\frac{a}{b}\right)=\boxed{\frac{a}{b}} if b ≠ 0 b\neq 0 since lim ⁡ t → 0 sin ⁡ t t = 1 \lim_{t\to 0}\frac{\sin{t}}{t}=1

1 Helpful 0 Interesting 0 Brilliant 0 Confused

won't the answer b/a be the same?

Harikrishna Nair - 5 years, 7 months ago

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b/a isn't the same as a/b unless a = b .

Otto Bretscher - 5 years, 7 months ago

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Sorry i made a small mistake............i took b/a outside instead of a/b............thanks. The solution is perfectly correct.

Harikrishna Nair - 5 years, 7 months ago

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