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Wize AP Calculus (AB) Textbook > Limits & Continuity

The Fundamental Trig Limit

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The Fundamental Trig Limit

The following identify is very useful in evaluating many limits in which our previous techniques won't work. This identity is sometimes known as the Fundamental Trigonometric Limit.
lim⁡θ→0 sin⁡θθ=1\boxed{\displaystyle \lim_{\theta \rightarrow0}\ \frac{\sin{\theta}}{\theta}=1}


Wize Tip
This works with the reciprocal as well! lim⁡θ→0 θsin⁡θ=1\boxed{\displaystyle \lim_{\theta \rightarrow0}\ \frac{\theta}{\sin{\theta}}=1}


A More General Version

We can use the Fundamental Trig Limit more generally, if lim⁡x→ag(x)=0\displaystyle \lim_{x\rightarrow a} g(x)=0 then

lim⁡x→a sin⁡(g(x))g(x)=1\boxed{\displaystyle \lim_{x\rightarrow a}\ \frac{\sin{(g(x)})}{g(x)}=1}

A Similar Identity

Another identity that also arises is


lim⁡θ→01−cos⁡θθ=0\boxed{\displaystyle \lim_{\theta\rightarrow0}\frac{1-\cos{\theta}}{\theta}=0}

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Example: The Fundamental Trig Limit

Find the following limit

lim⁡x→0sin⁡5xsin⁡3x\displaystyle \lim_{x\rightarrow0}\frac{\sin{5x}}{\sin{3x}}



lim⁡x→0sin⁡5xsin⁡3x=lim⁡x→0sin⁡5x11sin⁡3x=lim⁡x→05sin⁡5x5x3xsin⁡3x13=53lim⁡x→0sin⁡5x5x3xsin⁡3x=53∗1∗1=53\displaystyle \lim_{x\rightarrow0}\frac{\sin{5x}}{\sin{3x}}=\displaystyle \lim_{x\rightarrow0}\frac{\sin{5x}}{1} \frac{1}{\sin{3x}} \\ \text{} \\=\displaystyle \lim_{x\rightarrow0}5\frac{\sin{5x}}{5x} \frac{3x}{\sin{3x}}\frac{1}{3} \\ \text{} \\=\frac{5}{3}\displaystyle \lim_{x\rightarrow0}\frac{\sin{5x}}{5x} \frac{3x}{\sin{3x}}\\ \text{} \\=\frac{5}{3}*1*1\\ \text{} \\=\boxed{\frac{5}{3}}