Calculus

Limits and Continuity of Functions

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Trigonometric Limits

  • 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 \(\lim\limits_{x \to 0} {\large\frac{{\sin \left( {x – a} \right) – \sin \left( {x + a} \right)}}{x}\normalsize}.\)

    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

    \[{\lim\limits_{x \to 0} \left( {2\cos 4x} \right) }
    = {2\lim\limits_{x \to 0} \cos 4x }
    = {2 \cdot \cos \left( {4 \cdot 0} \right) = 2 \cdot 1 = 2.}
    \]

    Page 1
    Problems 1-3
    Page 2
    Problems 4-9