Question:

\(\lim_{{x \to 0}} \limits\) \(\frac{cos(sin x)  -  cos x }{x^4}\) is equal to :

Updated On: Sep 24, 2024
  • \(\frac{1}{3}\)

  • \(\frac{1}{4}\)

  • \(\frac{1}{6}\)

  • \(\frac{1}{12}\)

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The Correct Option is C

Solution and Explanation

The correct answer is (C) : \(\frac{-11}{9}\)

 \(\lim_{{x \to 0}} \limits\) \(\frac{cos(sinx) - cosx}{x^4}\) = \(\lim_{{x \to 0}} \limits\) \(\frac{2sin(x + sinx).sin(\frac{x - sinx}{2})}{x^4}\)
\(\lim_{{x \to 0}} \limits\) \(2.\frac{(\frac{( x + sinx }{2})(\frac{x-sinx}{2})}{x^4}\)
\(\lim_{{x \to 0}} \limits\) \(\frac{1}{2}.\) \((\frac{(x+x -\frac{x^3}{3!}+\frac{x^5}{5!}....)(x-x+\frac{x^3}{3!}....}{x^4})\)
\(\lim_{{x \to 0}} \limits\) \(\frac{1}{2}.\) \((2-\frac{x^2}{3!}+\frac{x^4}{5!}....)(\frac{1}{3!}-\frac{x^2}{5!}-1)\)
\(\frac{1}{6}\)

 

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Concepts Used:

Limits of Trigonometric Functions

Assume a is any number in the general domain of the corresponding trigonometric function, then we can explain the following limits.

Limits of Trigonometric Functions

We know that the graphs of the functions y = sin x and y = cos x detain distinct values between -1 and 1 as represented in the above figure. Thus, the function is swinging between the values, so it will be impossible for us to obtain the limit of y = sin x and y = cos x as x tends to ±∞. Hence, the limits of all six trigonometric functions when x tends to ±∞ are tabulated below: