\(\lim_{{x \to \frac{1}{\sqrt{2}}}} \frac{\sin(\cos^{-1}(x)) - x}{1 - \tan(\cos^{-1}(x))}\)
is equal to :
\(\lim_{{x \to \frac{1}{\sqrt{2}}}} \frac{\sin(\cos^{-1}(x)) - x}{1 - \tan(\cos^{-1}(x))}\)
is equal to :
\(\sqrt2\)
\(-\sqrt2\)
\(\frac{1}{\sqrt2}\)
\(-\frac{1}{\sqrt2}\)
The Correct Option is D
Solution and Explanation
The correct answer is (D) : \(-\frac{1}{\sqrt2}\)
\(\lim_{{x \to \frac{1}{\sqrt{2}}}} \frac{\sin(\cos^{-1}(x)) - x}{1 - \tan(\cos^{-1}(x))}\)
let \(cos^{−1}x=\frac{π}{4}+θ\)
\(\lim_{{\theta \to 0}} \frac{{\sin\left(\frac{\pi}{4} + \theta\right) - \cos\left(\frac{\pi}{4} + \theta\right)}}{{1 - \tan\left(\frac{\pi}{4} + \theta\right)}}\)
\(\lim_{{\theta \to 0}} \frac{{\sqrt{2}\sin\left(\frac{\pi}{4} + \theta - \frac{\pi}{4}\right)}}{{1 - \frac{1 + \tan\theta}{1 - \tan\theta}}}\)
\(\lim_{{\theta \to 0}} \frac{{\sqrt{2}\sin(\theta)}}{{-2\tan(\theta)}}(1 - \tan(\theta) = -\frac{1}{\sqrt{2}}\)
Top Questions on Limits
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Then:
Questions Asked in JEE Main exam
MX is a sparingly soluble salt that follows the given solubility equilibrium at 298 K.
MX(s) $\rightleftharpoons M^{+(aq) }+ X^{-}(aq)$; $K_{sp} = 10^{-10}$
If the standard reduction potential for $M^{+}(aq) + e^{-} \rightarrow M(s)$ is $(E^{\circ}_{M^{+}/M}) = 0.79$ V, then the value of the standard reduction potential for the metal/metal insoluble salt electrode $E^{\circ}_{X^{-}/MX(s)/M}$ is ____________ mV. (nearest integer)
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An infinitely long straight wire carrying current $I$ is bent in a planar shape as shown in the diagram. The radius of the circular part is $r$. The magnetic field at the centre $O$ of the circular loop is :
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- The hydrogen spectrum consists of several spectral lines in Lyman series (L$_1$, L$_2$, L$_3$ ...; L$_1$ has lowest energy among Lyman series). Similarly, it consists of several spectral lines in Balmer series (B$_1$, B$_2$, B$_3$ ...; B$_1$ has lowest energy among Balmer lines). The energy of L$_1$ is $x$ times the energy of B$_1$. The value of $x$ is ___ $\times 10^{-1}$. (Nearest integer)
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Concepts Used:
Limits
A function's limit is a number that a function reaches when its independent variable comes to a certain value. The value (say a) to which the function f(x) approaches casually as the independent variable x approaches casually a given value "A" denoted as f(x) = A.
If limx→a- f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the left of ‘a’. This value is also called the left-hand limit of ‘f’ at a.
If limx→a+ f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the right of ‘a’. This value is also called the right-hand limit of f(x) at a.
If the right-hand and left-hand limits concur, then it is referred to as a common value as the limit of f(x) at x = a and denote it by lim x→a f(x).