\(\lim_{{x \to \frac{1}{\sqrt{2}}}} \frac{\sin(\cos^{-1}(x)) - x}{1 - \tan(\cos^{-1}(x))}\)
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 \(cos^{−1}x=\frac{π}{4}+θ\)
\(\lim_{{\theta \to 0}} \frac{{\sin\left(\frac{\pi}{4} + \theta\right) - \cos\left(\frac{\pi}{4} + \theta\right)}}{{1 - \tan\left(\frac{\pi}{4} + \theta\right)}}\)
\(\lim_{{\theta \to 0}} \frac{{\sqrt{2}\sin\left(\frac{\pi}{4} + \theta - \frac{\pi}{4}\right)}}{{1 - \frac{1 + \tan\theta}{1 - \tan\theta}}}\)
\(\lim_{{\theta \to 0}} \frac{{\sqrt{2}\sin(\theta)}}{{-2\tan(\theta)}}(1 - \tan(\theta) = -\frac{1}{\sqrt{2}}\)
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Questions Asked in JEE Main exam
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If the domain of the function \( f(x) = \frac{1}{\sqrt{3x + 10 - x^2}} + \frac{1}{\sqrt{x + |x|}} \) is \( (a, b) \), then \( (1 + a)^2 + b^2 \) is equal to:
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- Let \( f: [0, 3] \to A \) be defined by \( f(x) = 2x^3 - 15x^2 + 36x + 7 \) and \( g: [0, \infty) \to B \) be defined by \( g(x) = \frac{x}{x^{2025} + 1}. \) If both functions are onto and \( S = \{ x \in \mathbb{Z} : x \in A { or } x \in B \} \), then \( n(S) \) is equal to:
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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).