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Wize University Calculus 1 Textbook > Limits

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}}











Extra Practice
Trig Limits
Find the value of lim⁡x→0sin⁡2xsin⁡3x\displaystyle \lim_{x\rightarrow0}\frac{\sin2x}{\sin3x}.
Fundamental trigonometric limits
Evaluate lim⁡x→0tan⁡x−sin⁡xxcos⁡x\lim\limits_{x\to0}\frac{\tan x-\sin x}{x\cos x}.
Fundamental trigonometric limits
Evaluate lim⁡x→0tan⁡x−sin⁡xxcos⁡x\lim\limits_{x\to0}\frac{\tan x-\sin x}{x\cos x}.
Fundamental trigonometric limits
Determine lim⁡x→0x2tan⁡2 2x\lim\limits_{x\rightarrow0}\frac{x^2}{\tan^2\ 2x}.
Trig Limits
Find the value of lim⁡x→0sin⁡2xsin⁡3x\displaystyle \lim_{x\rightarrow0}\frac{\sin2x}{\sin3x}.
Trigonometric Limits
Practice: Trig Limits
Evaluate lim⁡x→0 2sin⁡5xx2\displaystyle\lim_{x\to0}\ \frac{2\sin5x}{x^2}.
Trig Limits
Find the value of lim⁡x→0sin⁡2xsin⁡3x\displaystyle \lim_{x\rightarrow0}\frac{\sin2x}{\sin3x}.
Trigonometric Limits
Practice: Trig Limits
Evaluate lim⁡x→0 2sin⁡5xx2\displaystyle\lim_{x\to0}\ \frac{2\sin5x}{x^2}.
Practice: Using Known Trig Limits Indeterminate Forms
Behaviour of Trigonometric Functions: Evaluate lim⁡x→0sin⁡7x4x\lim_{x\to0}\limits\dfrac{\sin 7x}{4x}
Limits: Trigonometric Functions
Evaluate lim⁡x→3x2−2x−3πsin⁡(x−3)\displaystyle\lim_{x\to 3}\dfrac{x^2-2x-3}{\pi\sin(x-3)}
Limits: Trigonometric Functions
Evaluate lim⁡x→0tan⁡2x4x(x2−1)\displaystyle\lim_{x\to 0}\dfrac{\tan 2x}{4x(x^2-1)}
Evaluate lim⁡x→0sin⁡7x4x\displaystyle \lim_{x\rightarrow 0}\frac{\sin{7x}}{4x}.
Fundamental trigonometric limits
Evaluate lim⁡x→0tan⁡x−sin⁡xxcos⁡x\lim\limits_{x\to0}\frac{\tan x-\sin x}{x\cos x}.
Trigonometric Limits: Combining techniques
Evaluatelim⁡x→0xln⁡(1x+2x)e1+tan⁡ x+ln⁡ x(1+arctan⁡ x)sin⁡ x+xcos⁡ x\lim\limits_{x\rightarrow0}\frac{x\ln(\frac{1}{x+2^x})e^{{1+\tan\ x+\ln\ x}}}{(1+\arctan\ x)\sin\ x+x\cos\ x}
🌶️ TOUGH! Evaluate lim⁡x→πsin⁡xx−π\lim\limits_{x\rightarrow \pi}\frac{\sin x}{x-\pi}. Hint: use the trigonometric identity sin⁡(θ+π)=−sin⁡(θ)\sin(\theta+\pi)=-\sin(\theta).
Determine lim⁡x→0x2tan⁡23x\lim\limits_{x\rightarrow 0}\frac{x^2}{\tan^23x}.
Evaluate the following limit.
lim⁡x→0tan⁡(ax)sin⁡x\displaystyle \lim_{x\rightarrow 0}\frac{\tan(ax)}{\sin x}
Consider f(x)=sin⁡2xx\displaystyle f(x)=\frac{\sin{2x}}{x} . Does lim⁡x→0sin⁡2xx\displaystyle \lim_{x\rightarrow 0}\frac{\sin{2x}}{x} exist? Is ff continuous at x=0x=0?
Practice: Fundamental trigonometric limits
Evaluatelim⁡x→1−sin⁡1−x1−x2\lim\limits_{x\rightarrow1^-}\frac{\sin\sqrt{1-x}}{\sqrt{1-x^2}}
Practice: Fundamental trigonometric limits
Part a)
Evaluatelim⁡x→0sin⁡(2πx)sin⁡(3πx)\lim\limits_{x\rightarrow0}\frac{\sin(2\pi x)}{\sin(3\pi x)}
Part b)
Fundamental trigonometric limits
Determine lim⁡x→0x2tan⁡2 2x\lim\limits_{x\rightarrow0}\frac{x^2}{\tan^2\ 2x}.
Trig Limits
Find the value of lim⁡x→0sin⁡2xsin⁡3x\displaystyle \lim_{x\rightarrow0}\frac{\sin2x}{\sin3x}.
Trigonometric Limits
Practice: Trig Limits
Evaluate lim⁡x→0 2sin⁡5xx2\displaystyle\lim_{x\to0}\ \frac{2\sin5x}{x^2}.
Fundamental Trig Limit
Evaluate the following limit:
lim⁡x→0sin⁡(3x)cos⁡(2x)sin⁡(5x)\lim_{x\rightarrow0}\frac{\sin(3x)\cos(2x)}{\sin(5x)}
Each question is worth 3 marks.
Limits and Continuity
Evaluate the following limits:
a) lim⁡x→4x−4x−4\displaystyle \lim_{x \rightarrow 4} \frac{\sqrt{x} - \sqrt{4}}{x - 4}
b) lim⁡x→∞x2−5x+134x4+3x−21\displaystyle \lim_{x \rightarrow \infty} \frac{x^2 - 5x + 13}{\sqrt{4x^4 +3x - 21}}