MHT CET · Maths · Continuity and Differentiability
If \(\begin{aligned} f(x) &=\frac{4 \sin \pi x}{5 x} \text { for } x \neq 0 \ &=2 \mathrm{k} \quad \text { for } x=0 \end{aligned}\)
is continuous at \(x=0\), then the value of \(k\) is
- A \(\frac{2 \pi}{5}\)
- B \(\frac{\pi}{5}\)
- C \(\frac{\pi}{10}\)
- D \(\frac{4 \pi}{5}\)
Answer & Solution
Correct Answer
(A) \(\frac{2 \pi}{5}\)
Step-by-step Solution
Detailed explanation
(C)
Since \(f(x)\) is continuous at \(x=0\),
\(\begin{array}{l}
\lim _{x \rightarrow 0} f(x)=f(0) \Rightarrow \lim _{x \rightarrow 0} \frac{4 \sin \pi x}{5 x}=2 k \\
\therefore \lim _{x \rightarrow 0}\left(\frac{4 \sin \pi x}{\pi x}\right) \cdot \frac{\pi}{5}=2 k \Rightarrow(1) \cdot \frac{4 \pi}{5}=2 k \Rightarrow k=\frac{2 \pi}{5}
\end{array}\)
Since \(f(x)\) is continuous at \(x=0\),
\(\begin{array}{l}
\lim _{x \rightarrow 0} f(x)=f(0) \Rightarrow \lim _{x \rightarrow 0} \frac{4 \sin \pi x}{5 x}=2 k \\
\therefore \lim _{x \rightarrow 0}\left(\frac{4 \sin \pi x}{\pi x}\right) \cdot \frac{\pi}{5}=2 k \Rightarrow(1) \cdot \frac{4 \pi}{5}=2 k \Rightarrow k=\frac{2 \pi}{5}
\end{array}\)
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 general solutions of \(\sin ^{2} x \cdot \sec x=\tan x-\sin x+1\) isMHT CET 2020 Medium
- The statement pattern \([(p \rightarrow q) \wedge \sim q] \rightarrow r\) is a tautology when \(r\) is equivalent toMHT CET 2025 Medium
- The order of the differential equation whose solution is \(y=a \cos x+b \sin x+c e^{-x}\), isMHT CET 2007 Easy
- Let \(\overline{\mathrm{a}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}\) and \(\overline{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}\). If \(\overline{\mathrm{c}}\) is a vector such that \(\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=|\overline{\mathrm{c}}|,|\overline{\mathrm{c}}-\overline{\mathrm{a}}|=2 \sqrt{2}\) and the angle between \((\overline{\mathrm{a}} \times \overline{\mathrm{b}})\) and \(\overline{\mathrm{c}}\) is \(\frac{\pi}{6}\), then \(|(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times \overline{\mathrm{c}}|\) isMHT CET 2023 Hard
- The point on the curve where the tangent is perpendicular to the line isMHT CET 2017 Medium
- If \(\bar{a}=2 \hat{i}+3 \hat{j}-\hat{k}, \bar{b}=-\hat{i}+2 \hat{j}-4 \hat{k}\) and \(\bar{c}=\hat{i}+\hat{j}-2 \hat{k}\), then \((\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \cdot(\overline{\mathrm{a}} \times \overline{\mathrm{c}})=\)MHT CET 2021 Easy
More PYQs from MHT CET
- In an isobaric process of an ideal gas, the ratio of heat supplied and work done by the system \(\left(\frac{\mathrm{Q}}{\mathrm{W}}\right)\) is \(\left[\frac{\mathrm{C}_{\mathrm{P}}}{\mathrm{C}_{\mathrm{V}}}=\gamma\right]\)MHT CET 2025 Medium
- Calculate \(\Delta S\) total for a certain reaction at 298 K if \(\Delta \mathrm{H}^{\circ}=-208.6 \mathrm{~kJ}\) and \(\Delta S^{\circ}=-36 J^{-1}\)MHT CET 2025 Medium
- Which from following statements is CORRECT about saccharic acid?MHT CET 2024 Medium
- Identify the correct set of labelling in the given diagram.
MHT CET 2020 Hard - For the system \(x-y+z=4,2 x+y-3 z=0\), \(x+y+z=2\), the values of \(x, y, z\) respectively are given byMHT CET 2024 Medium
- A plane makes positive intercepts of unit length on each of \(X\) and \(Y\) axis. If it passes through the point \((-1,1,2)\) and makes angle \(\theta\) with the X -axis, then \(\theta\) isMHT CET 2024 Medium