JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Let \(f(x)=\lim _{\theta \rightarrow 0}\left(\frac{\cos \pi x-x^{\left(\frac{2}{\theta}\right)} \sin (x-1)}{1+x^{\left(\frac{2}{\theta}\right)}(x-1)}\right), x \in R\).
Consider the following two statements :
(I) \(f ( x )\) is discontinous at \(x =1\).
(II) \(f ( x )\) is continous at \(x =-1\). Then,
- A Neither (I) nor (II) is True
- B Both (I) and (II) are True
- C Only (II) is True
- D Only (I) is True
Answer & Solution
Correct Answer
(A) Neither (I) nor (II) is True
Step-by-step Solution
Detailed explanation
\(f(x)=\left\{\begin{array}{cc}\cos \pi x & x \rightarrow 1^{-} \\ \frac{-\sin (x-1)}{(x-1)} & x \rightarrow 1^{+}\end{array}\right.\) \(RHL =\lim _{ x \rightarrow 1} \frac{-\sin ( x -1)}{( x -1)}=-1\) \(LHL =\lim _{ x \rightarrow 1} \cos \pi x =-1, f (1)=-1\) \(f ( x )\) is…
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 term independent of \(x\) in expansion of \({\left( {\frac{{x + 1}}{{{x^{2/3}} - {x^{\frac{1}{3}}} + 1\;}}--\frac{{x - 1}}{{x - {x^{1/2}}}}} \right)^{10}}\) isJEE Mains 2013 Hard
- Let a line \(l\) pass through the origin and be perpendicular to the lines \(l_1: \overrightarrow{ r }=(\hat{ i }-11 \hat{ j }-7 \hat{ k })+\lambda(\hat{ i }+2 \hat{ j }+3 \hat{ k }), \lambda \in R\) and \(l_2: \overrightarrow{ r }=(-\hat{ i }+\hat{ k })+\mu(2 \hat{ i }+2 \hat{ j }+\hat{ k }), \mu \in R\). If \(P\) is the point of intersection of \(l\) and \(l_1\), and \(Q (\alpha\) \(, \beta, \gamma)\) is the foot of perpendicular from \(P\) on \(l_2\), then \(9(\alpha+\beta+\gamma)\) is equal to \(..........\).JEE Mains 2023 Hard
- The distance of the point \((7,-3,-4)\) from the plane passing through the points \((2,-3,1),(-1,1,-2)\) and \((3,-4,2)\) is:JEE Mains 2023 Easy
- The number of distinct real solutions of the equation \(x|x+4|+3|x+2|+10=0\) isJEE Mains 2026 Hard
- An angle of intersection of the curves, \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) and \(\mathrm{x}^{2}+\mathrm{y}^{2}=\mathrm{ab}, \mathrm{a}>\mathrm{b}\), is :JEE Mains 2021 Hard
- The number of critical points of the function \(f(x)=(x-2)^{2 / 3}(2 x+1)\) is :JEE Mains 2024 Hard
More PYQs from JEE Mains
- Let the sum of the maximum and the minimum values of the function \(f(x)=\frac{2 x^2-3 x+8}{2 x^2+3 x+8}\) be \(\frac{m}{n}\), where \(\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1\). Then \(\mathrm{m}+\mathrm{n}\) is equal to :JEE Mains 2024 Hard
- Let the maximum and minimum values of \(\left(\sqrt{8 x-x^2-12}-4\right)^2+(x-7)^2, x \in R\) be \(M\) and \(m\) respectively. Then \(\mathrm{M}^2-\mathrm{m}^2\) is equal to ...............JEE Mains 2024 Hard
- Let \(\mathrm{A}(-2,-1), \mathrm{B}(1,0), \mathrm{C}(\alpha, \beta)\) and \(\mathrm{D}(\gamma, \delta)\) be the vertices of a parallelogram \(A B C D\). If the point \(C\) lies on \(2 x-y=5\) and the point \(D\) lies on \(3 x-2 y=6\), then the value of \(|\alpha+\beta+\gamma+\delta|\) is equal to ...........JEE Mains 2024 Hard
- If \(A\) and \(B\) are two events such that \(P(A \cap B)=0.1\), and \(P(A \mid B)\) and \(P(B \mid A)\) are the roots of the equation \(12 x^2-7 x+1=0\), then the value of \(\frac{\mathrm{P}(\overline{\mathrm{A}} \cup \overline{\mathrm{B}})}{\mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}})}\) is :JEE Mains 2025 Hard
- Let \(A=\{0,3,4,6,7,8,9,10\} \quad\) and \(R\) be the relation defined on A such that \(R =\{( x , y ) \in A \times A : x - y \quad\) is odd positive integer or \(x-y=2\}\). The minimum number of elements that must be added to the relation \(R\), so that it is a symmetric relation, is equal to \(...........\).JEE Mains 2023 Hard
- Let \(y=f(x)\) be the solution of the differential equation \(y(x+1) d x-x^2 d y=0, y(1)=e\). Then \(\lim _{x \rightarrow 0^{+}} f(x)\) is equal toJEE Mains 2023 Hard