JEE Mains · Maths · STD 12 - 5. continuity and differentiation
If \(f\left( x \right)\left\{ {\begin{array}{*{20}{c}}
{\frac{{\sin \,\left( {p + 1} \right)x + \sin \,x}}{x},\,\,}&{x < 0} \\
{q\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,}&{x = 0} \\
{\frac{{\sqrt {x + {x^2}} - \sqrt x }}{{x/2}},}&{x > 0}
\end{array}} \right.\) Is continuous at \(x = 0\), then the ordered pair \((p, q)\) is equal to
- A \(\left( { - \frac{3}{2}, - \frac{1}{2}} \right)\)
- B \(\left( {\frac{5}{2},\frac{1}{2}} \right)\)
- C \(\left( { - \frac{1}{2},\frac{3}{2}} \right)\)
- D \(\left( { - \frac{3}{2}, \frac{1}{2}} \right)\)
Answer & Solution
Correct Answer
(D) \(\left( { - \frac{3}{2}, \frac{1}{2}} \right)\)
Step-by-step Solution
Detailed explanation
\(RHL = \mathop {\lim }\limits_{x \to {0^ + }} \frac{{\sqrt {x + {x^2}} - \sqrt x }}{{{x^{3/2}}}}\) \( = \mathop {\lim }\limits_{x \to {0^ + }} \frac{{\sqrt {1 + x} - 1}}{x} = \frac{1}{2}\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- The number of integers greater than \(6000\) that can be formed , using the digits \(3,5,6,7,\) and \( 8\) without repetition is :JEE Mains 2014 Hard
- The eccentricity of an ellipse \(E\) with centre at the origin \(O\) is \(\dfrac{\sqrt{3}}{2}\) and its directrices are \(x = \pm \dfrac{4\sqrt{6}}{3}\). Let \(H: \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1\) be a hyperbola whose eccentricity is equal to the length of semi-major axis of \(E\), and whose length of latus rectum is equal to the length of minor axis of \(E\). Then the distance between the foci of \(H\) is :JEE Mains 2026 Hard
- A symmetrical form of the line of intersection of the planes \(x = ay + b\) and \(z = cy + d\) isJEE Mains 2014 Medium
- The value of \(\int \limits_{-\pi / 2}^{\pi / 2} \frac{1}{1+ e ^{\sin x}} d x\)JEE Mains 2020 Medium
- If the set of all solutions of \(|x^2 + x - 9| = |x| + |x^2 - 9|\) is \([\alpha, \beta] \cup [\gamma, \infty)\), then \((\alpha^2 + \beta^2 + \gamma^2)\) is equal to:JEE Mains 2026 Medium
- Consider the function \(f : R \rightarrow R\) defined by \(f(x)=\left\{\begin{array}{cc}\left(2-\sin \left(\frac{1}{x}\right)\right)|x|, x \neq 0 \\ 0 & , x=0\end{array} .\right.\) Then \(f\) isJEE Mains 2021 Hard
More PYQs from JEE Mains
- An angle between the plane, \(x + y + z = 5\) and the line of intersection of the planes, \(3x + 4y + z- 1 = 0\) and \(5x + 8y + 2z+ 14 = 0\) , isJEE Mains 2018 Hard
- Let \(S=\left\{x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right): 9^{1-\tan ^2 x}+9^{\tan ^2 x}=10\right\}\) and \(\beta=\sum_{x \in S} \tan ^2\left(\frac{x}{3}\right)\), then \(\frac{1}{6}(\beta-14)^2\) is equal toJEE Mains 2023 Hard
- Let the relations \(R_1\) and \(R_2\) on the set \(\mathrm{X}=\{1,2,3, \ldots, 20\}\) be given by \(\mathrm{R}_1=\{(\mathrm{x}, \mathrm{y}): 2 \mathrm{x}-3 \mathrm{y}=2\}\) and \(\mathrm{R}_2=\{(\mathrm{x}, \mathrm{y}):-5 \mathrm{x}+4 \mathrm{y}=0\}\). If \(\mathrm{M}\) and \(\mathrm{N}\) be the minimum number of elements required to be added in \(R_1\) and \(R_2\), respectively, in order to make the relations symmetric, then \(\mathrm{M}+\mathrm{N}\) equalsJEE Mains 2024 Hard
- A line in the \(3-\) dimensional space makes an angle \(\theta \left( {0 < \theta \le \frac{\pi }{2}} \right)\) with both the \(x\) and \(y\) axes. Then the set ofall values of \(\theta \) is the intervalJEE Mains 2014 Hard
- Let \(\alpha_\theta\) and \(\beta_\theta\) be the distinct roots of \(2 x^2+(\cos \theta) x-1=0, \theta \in(0,2 \pi)\). If m and M are the minimum and the maximum values of \(\alpha_\theta^4+\beta_\theta^4\), then \(16(M+m)\) equals :JEE Mains 2025 Medium
- If \(a+\alpha=1, b+\beta=2\) and \(\operatorname{af}(x)+\alpha f\left(\frac{1}{x}\right)=b x+\frac{\beta}{x}, x \neq 0,\) then the value of expression \(\frac{ f ( x )+ f \left(\frac{1}{ x }\right)}{ x +\frac{1}{ x }}\) is ..... .JEE Mains 2021 Hard