Math Doubts

Limits of Trigonometric functions

Let a literal $x$ denotes an angle of right triangle. Then, the trigonometric functions sine, cosine, tangent, cotangent, secant and cosecant are written as $\sin{x}$, $\cos{x}$, $\tan{x}$, $\cot{x}$, $\sec{x}$ and $\csc{x}$ respectively. Now, let’s learn the limits of trigonometric functions with proofs.

Sine

$(1).\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \sin{x}}$ $\,=\,$ $0$

$(2).\,\,$ $\displaystyle \large \lim_{x\,\to\,\pm \infty}{\normalsize \sin{x}}$ $\,=\,$ $Undefined$

Cosine

$(1).\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \cos{x}}$ $\,=\,$ $1$

$(2).\,\,$ $\displaystyle \large \lim_{x\,\to\,\pm \infty}{\normalsize \cos{x}}$ $\,=\,$ $Undefined$

Tangent

$(1).\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \tan{x}}$ $\,=\,$ $0$

$(2).\,\,$ $\displaystyle \large \lim_{x\,\to\,\pm \infty}{\normalsize \tan{x}}$ $\,=\,$ $Undefined$

Co-Tangent

$(1).\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \cot{x}}$ $\,=\,$ $\infty$

$(2).\,\,$ $\displaystyle \large \lim_{x\,\to\,\pm \infty}{\normalsize \cot{x}}$ $\,=\,$ $Undefined$

Secant

$(1).\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \sec{x}}$ $\,=\,$ $1$

$(2).\,\,$ $\displaystyle \large \lim_{x\,\to\,\pm \infty}{\normalsize \sec{x}}$ $\,=\,$ $Undefined$

Cosecant

$(1).\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \csc{x}}$ $\,=\,$ $\infty$

$(2).\,\,$ $\displaystyle \large \lim_{x\,\to\,\pm \infty}{\normalsize \csc{x}}$ $\,=\,$ $Undefined$

Formulas

$(1).\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\sin{x}}{x}}$ $\,=\,$ $1$

$(2).\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\tan{x}}{x}}$ $\,=\,$ $1$

There are two standard limit formulas with trigonometric functions in calculus and examples to learn how to use them in finding the limits of trigonometric functions.

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Worksheet

$(1).\,\,$ $\displaystyle \large \lim_{x \,\to\, 1}{\normalsize \dfrac{\sin{(x-1)}}{x^2-1}}$

$(2).\,\,$ $\large \displaystyle \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin{(\pi\cos^2{x})}}{x^2}}$

$(3).\,\,$ $\displaystyle \large \lim_{x \,\to\, 0} \normalsize \dfrac{1-\cos{6x}}{1-\cos{7x}}$

The list of limits questions with solutions for practice and to learn how to find the limits of functions in which the trigonometric functions are involved.

Solutions