Maths · Limits, continuity and differentiability
If =\left|\begin{array}{ccc}\cos x & x & 1 \\ 2 \sin x & x^{2} & 2 x \\ \tan x &
If \( f(x)=\left|\begin{array}{ccc}\cos x & x & 1 \\ 2 \sin x & x^{2} & 2 x \\ \tan x & x & 1\end{array}\right|, \) then \( \lim _{x \rightarrow 0} \frac{f^{\prime}(x)}{x} \)
- A. Exists and is equal to -2
- B. Does not exist
- C. Exist and is equal to 0
- D. Exists and is equal to 2
Step-by-step solution
Compute the determinant to get f(x)=x^2(tan x - cos x). Differentiate: f'(x)=2x(tan x - cos x) + x^2(sec^2 x + sin x). Then lim_{x→0} f'(x)/x = lim_{x→0} [2 tan x - 2 cos x + x sec^2 x + x sin x] = 2·0 - 2·1 + 0 + 0 = -2. Hence the limit exists and equals -2.
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