The basic trigonometric limit is
\[\lim\limits_{x \to 0} \frac{{\sin x}}{x} = 1.\]
Using this limit, one can get the series of other trigonometric limits:
\[{\lim\limits_{x \to 0} \frac{{\tan x}}{x} = 1,\;\;\;}\kern-0.3pt
{\lim\limits_{x \to 0} \frac{{\arcsin x}}{x} = 1,\;\;\;}\kern-0.3pt
{\lim\limits_{x \to 0} \frac{{\arctan x}}{x} = 1.}\]
Further we assume that angles are measured in radians.
Solved Problems
Click a problem to see the solution.
Example 1
Find the limit \(\lim\limits_{x \to 0} {\large{\frac{{4x}}{{\sin 3x}}}\normalsize}\).Example 2
Calculate the limit \(\lim\limits_{x \to 0} {\large\frac{{\cos {3x} – \cos x}}{{{x^2}}}\normalsize}\).Example 3
Find the limit \(\lim\limits_{x \to 0} {\large\frac{{\sin5x – \sin 3x}}{{\sin x}}\normalsize}\).Example 4
Calculate the limit \(\lim\limits_{x \to 0} {\large{\frac{{\cos \left( {x + a} \right) – \cos \left( {x – a} \right)}}{x}}\normalsize}.\)Example 5
Calculate the limit \(\lim\limits_{x \to 0} {\large\frac{{\sin ax}}{{\sin bx}}\normalsize}\).Example 6
Find the limit \(\lim\limits_{x \to b} {\large\frac{{\sin x – \sin b}}{{x – b}}\normalsize}\).Example 7
Find the limit \(\lim\limits_{x \to 0} {\large\frac{{\tan x – \sin x}}{{{x^3}}}\normalsize}\).Example 8
Find the limit \(\lim\limits_{x \to {\large\frac{1}{2}\normalsize}} {\large\frac{{1 – 4{x^2}}}{{\arcsin \left( {1 – 2x} \right)}}\normalsize}\).Example 9
Find the limit \(\lim\limits_{x \to 0 + 0} {\large\frac{{\sqrt {1 – \cos x} }}{x}\normalsize}\).Example 1.
Find the limit \(\lim\limits_{x \to 0} {\large{\frac{{4x}}{{\sin 3x}}}\normalsize}\).Solution.
\[L
= {\lim\limits_{x \to 0} \frac{{4x}}{{\sin 3x}} }
= {\lim\limits_{x \to 0} \frac{{3 \cdot 4x}}{{3\sin 3x}} }
= {\frac{4}{3}\lim\limits_{x \to 0} \frac{{3x}}{{\sin 3x}} }
= {\frac{4}{3}\lim\limits_{x \to 0} \frac{1}{{\large\frac{{\sin 3x}}{{3x}}\normalsize}} }
= {\frac{4}{3}\frac{{\lim\limits_{x \to 0} 1}}{{\lim\limits_{x \to 0} \large\frac{{\sin 3x}}{{3x}}\normalsize}}.}
\]
Since \(3x \to 0\) as \(x \to 0,\) we can write:
\[L
= \frac{4}{3}\frac{{\lim\limits_{x \to 0} 1}}{{\lim\limits_{x \to 0} \large\frac{{\sin 3x}}{{3x}}\normalsize}}
= \frac{4}{{3\lim\limits_{3x \to 0} \large\frac{{\sin 3x}}{{3x}}\normalsize}}
= \frac{4}{{3 \cdot 1}} = \frac{4}{3}.
\]
Example 2.
Calculate the limit \(\lim\limits_{x \to 0} {\large\frac{{\cos {3x} – \cos x}}{{{x^2}}}\normalsize}\).Solution.
We factor the numerator:
\[{\cos{3x} – \cos x }
= { – 2\sin \frac{{3x – x}}{2}\sin \frac{{3x + x}}{2} }
= { – 2\sin x\sin 2x. }
\]
This yields
\[{\lim\limits_{x \to 0} \frac{{\cos 3x – \cos x}}{{{x^2}}} }
= {\lim\limits_{x \to 0} \frac{{\left( { – 2\sin x\sin 2x} \right)}}{{{x^2}}} }
= {- 2\lim\limits_{x \to 0} \frac{{\sin x}}{x} \cdot \lim\limits_{x \to 0} \frac{{\sin 2x}}{x} }
= {- 2 \cdot 1 \cdot \lim\limits_{2x \to 0} \frac{{2\sin 2x}}{{2x}} }
= {- 2 \cdot 2\lim\limits_{2x \to 0} \frac{{\sin 2x}}{{2x}} = – 4.}
\]
Example 3.
Find the limit \(\lim\limits_{x \to 0} {\large\frac{{\sin5x – \sin 3x}}{{\sin x}}\normalsize}\).Solution.
We use the following trigonometric identity:
\[{\sin x – \sin y }={ 2\sin \frac{{x – y}}{2}\cos \frac{{x + y}}{2}.}\]
Then we obtain
\[{\lim\limits_{x \to 0} \frac{{\sin5x – \sin 3x}}{{\sin x}} }
= {\lim\limits_{x \to 0} \frac{{2\sin \large\frac{{5x – 3x}}{2}\normalsize\cos \large\frac{{5x + 3x}}{2}\normalsize}}{{\sin x}} }
= {\lim\limits_{x \to 0} \frac{{2\sin x\cos 4x}}{{\sin x}} }
= {\lim\limits_{x \to 0} \left( {2\cos 4x} \right).}
\]
As \(\cos{4x}\) is a continuous function at \(x = 0,\) then
\[{\lim\limits_{x \to 0} \left( {2\cos 4x} \right) }
= {2\lim\limits_{x \to 0} \cos 4x }
= {2 \cdot \cos \left( {4 \cdot 0} \right) = 2 \cdot 1 = 2.}
\]