If \(f(x)\) is the integral of \(\dfrac{1+\cos^{2}x-2\cos x}{x\sin^{2}x\tan x}\), then what is \(\displaystyle\lim_{x\to 0}f'(x)\) equal to?
\(1/4\)
The numerator \(1+\cos^{2}x-2\cos x=(1-\cos x)^{2}\), so \(f'(x)=\dfrac{(1-\cos x)^{2}}{x\sin^{2}x\tan x}\). As \(x\to 0\), using \(1-\cos x\sim \dfrac{x^{2}}{2}\), \(\sin x\sim x\) and \(\tan x\sim x\), the expression behaves like \(\dfrac{x^{4}/4}{x^{4}}=\dfrac{1}{4}\), so the limit is \(1/4\).
Similar Questions
What is \(\rm \displaystyle\lim_{x\rightarrow 0} \dfrac{\sin x \log (1-x)}{x^2}\) equal to?
If a differentiable function f(x) satisfies \(\mathop {\lim }\limits_{x \to - 1} \dfrac{f(x)+1}{x^2-1}=-\dfrac{3}{2}\) then what is \(\mathop {\lim }\limits_{x \to - 1} f(x)\) equal to?
What is the value of \(\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin x{}^\circ }{\tan 3x{}^\circ }\) ?
What is \(\mathop {\lim }\limits_{x \to \frac{\pi }{6}} \;\frac{{2{{\sin }^2}x\; + {\rm{\;}}\sin x\; - {\rm{\;}}1}}{{2{{\sin }^2}x\; - {\rm{\;}}3\sin x\; + {\rm{\;}}1}}\) equal to?
What is \(\rm\mathop {\lim }\limits_{x \to 2}\frac{x^3 + x^2}{x^2 + 3x + 2}\) equal to?
If \(\rm\mathop {\lim }\limits_{x \to a} \frac{a^x -x^a}{x^x -a^a}= - 1\) , then what is the value of a?
What is \(\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 0} {\rm{\;}}{{\rm{e}}^{ - \frac{1}{{{{\rm{x}}^2}}}}}\) equal to?
What is \(\lim\limits_{x \to 0}x^3(\text{cosec} \ x)^2\) equal to ?
What is \(\rm \displaystyle\lim_{x \rightarrow 0} \dfrac{3^x + 3^{-x}-2}{x}\) equal to ?
If \(\rm \displaystyle\lim_{x \rightarrow 1} \dfrac{x^4-1}{x-1} = \lim_{x\rightarrow k} \dfrac{x^3-k^3}{x^2-k^2}\) , where k ≠ 0, then what is the value of k?
Important Questions from Evaluation of Limits
If $log 2 = 0.3010$ and $log 3 = 0.4771$, then the value of $log 36$ is
The L. C. M. of x2 - y2, x3 - y3 and x3 - x2y - xy2 + y3 is:
The series \(1 + \frac{2}{3} + {\left( {\frac{2}{3}} \right)^2} + ... + {\left( {\frac{2}{3}} \right)^{n - 1}}\) is:
The series \(\sum {\left( {\frac{1}{{np}}} \right)} \) is divergent if
If \(x + \frac{1}{x} = \sqrt{3}\), then the value of x18 + x12 + x6 + 1 is
