Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Continuity and Differentiability

NTA Abhyas 2020 · Mathematics · Continuity and Differentiability

If f(x) = \ cases x 4 + x - a & x > 0 c & x = 0 (4e(2)/(x) +...

If f\left(x\right) = \left\{\begin{cases}\dfrac{\sqrt{x}}{\sqrt{4 + \sqrt{x}} - a} & x > 0 \\ c & x = 0 \\ \dfrac{4e\dfrac{2}{x} + 3e\dfrac{1}{x}}{e\dfrac{2}{x} + be\dfrac{1}{x}} & x < 0\end{cases}\right) is continuous atx = 0forsomeconstantsfor some constantsa, b and cc, then the value of 50ba\dfrac{50b}{a} is equal to

Correct answer

18.75

Step-by-step solution

In lim⁡x→0+x4+x−a\lim_{x \to {0}^{+}}\dfrac{\sqrt{x}}{\sqrt{4 + \sqrt{x}} - a} for (00)\left(\dfrac{0}{0}\right) form a=2a = 2
⇒lim⁡x→0+x(4+x−2)×(4+x+2)(4+x+2)⇒ \lim_{x \to {0}^{+}}\dfrac{\sqrt{x}}{\left(\sqrt{4 + \sqrt{x}} - 2\right)} \times \dfrac{\left(\sqrt{4 + \sqrt{x}} + 2\right)}{\left(\sqrt{4 + \sqrt{x}} + 2\right)}
⇒lim⁡x→0+x(4+x+2)4+x−4=lim⁡x→0+4+x+2=4⇒ \lim_{x \to {0}^{+}}\dfrac{\sqrt{x}\left(\sqrt{4 + \sqrt{x}} + 2\right)}{4 + \sqrt{x} - 4} = \lim_{x \to {0}^{+}}\sqrt{4 + \sqrt{x}} + 2 = 4
Now, lim⁡x→0−f(x)=lim⁡x→0−4e2x+3e1xe2x+be1x\lim_{x \to {0}^{-}}f\left(x\right) = \lim_{x \to {0}^{-}}\dfrac{4{e}^{\dfrac{2}{x}} + 3{e}^{\dfrac{1}{x}}}{{e}^{\dfrac{2}{x}} + b{e}^{\dfrac{1}{x}}}
=lim⁡x→0+e1x(4e1x+3)e1x(e1x+b)=lim⁡x→0+4e1x+3e1x+b=0+30+b= \lim_{x \to {0}^{+}}\dfrac{{e}^{\dfrac{1}{x}}\left(4{e}^{\dfrac{1}{x}} + 3\right)}{{e}^{\dfrac{1}{x}}\left({e}^{\dfrac{1}{x}} + b\right)} = \lim_{x \to {0}^{+}}\dfrac{4{e}^{\dfrac{1}{x}} + 3}{{e}^{\dfrac{1}{x}} + b} = \dfrac{0 + 3}{0 + b}
For f(x)f\left(x\right) to be continuous at x=0x = 0
4=3b⇒b=344 = \dfrac{3}{b} ⇒ b = \dfrac{3}{4}
Practice Continuity and Differentiability on Quantrex Academy →

More from Continuity and Differentiability

Let ℝ denote the set of all real numbers. Let f : ℝ arrow ℝ be an arbitrary function and… JEE Advanced 2026Consider the function f : (-(π)/(2), (π)/(2)) arrow (-∞, ∞) defined by f(x) = (|x| + |x… JEE Advanced 2026For a real number α , let [α] denote the greatest integer less than or equal to α . For a… JEE Advanced 2026For the function f(x) = e^ |x| - |x| , x ∈ ℝ , consider the following statements… JEE Main 2026Let f(x) = cases (1)/(3), & x ≤ π/2 (b(1- x))/((π-2x)^2), & x > π/2 cases . If f is… JEE Main 2026Let f(x) = cases x^3 + 8 ; & x Then the number of points, where the function g ∘ f is… JEE Main 2026Let f(x)= cases e^ x-1 , & x<0 x^2-5x+6, & x ≥ 0 cases and g(x)=f(|x|)+|f(x)| . If the… JEE Main 2026The number of points, at which the function f(x) = \ 6x, 2 + 3x^2\ + |x - 1|| |x^2… JEE Main 2026The number of points in the interval [2, 4] , at which the function f(x) = [x^2 - x… JEE Main 2026Let f(x)= _ θ arrow 0 ( π x-x^ ((2)/(θ)) (x-1) 1+x^ ((2)/(θ)) (x-1) ), x ∈ R . Consider… JEE Main 2026Let [t] denote the greatest integer less than or equal to t . If the function f(x)=\… JEE Main 2026If the function f(x)= e^ x (e^ x-x -1)+ _ e ( x+ x)-x x-x is continuous at x=0 , then the… JEE Main 2026Let α, β ∈ ℝ be such that the function f(x)= cases 2 α(x^ 2 -2)+2 β x &, x be… JEE Main 2026If f(x)=\ array cc a|x|+x^ 2 -2( |x|)( |x|) x &, x ≠ 0 b &, x=0 array . is continuous at… JEE Main 2026Let f(x)= cases a x^ 2 +2 a x+3 4 x^ 2 +4 x-3 &, x ≠-(3)/(2), (1)/(2) ~b &, x=-(3)/(2)… JEE Main 2026Let [·] denote the greatest integer function, and let f(x)= \ 2 x, x^ 2 \ . Let S = \ x… JEE Main 2026 Full Continuity and Differentiability list Continuity and Differentiability JEE Maths PYQsJEE Mathematics PYQs All NTA Abhyas JEE Main PYQs