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Fundamental trigonometric limits

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Wize University Calculus 1 Textbook > Limits

The Fundamental Trig Limit

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Evaluate lim⁡x→0tan⁡x−sin⁡xxcos⁡x\lim\limits_{x\to0}\frac{\tan x-\sin x}{x\cos x}.
More The Fundamental Trig Limit Questions:
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
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}}