Exams › JEE Main › Maths

Let f(x)=lim_(x→0) ((cos πx − x^(2/5) sin(x−1)) / (1 + x^(2/5)(x−1))), x∈R. Consider the following two statements : (I) f(x) is discontinuous at x = 1. (II) f(x) is continuous at x = −1. Then,

1. Neither (I) nor (II) is True
2. Both (I) and (II) are True
3. Only (II) is True
4. Only (I) is True

Correct answer: Neither (I) nor (II) is True

Solution

The limit of the function as x approaches 0 does not indicate any discontinuity at x = 1, and the function remains continuous at x = -1, confirming that both statements are false.

Related JEE Main Maths questions

• Suppose $f(x)=\alpha(x)\beta(x)\gamma(x)$ for every real $x$, where $\alpha(x)$, $\beta(x)$, and $\gamma(x)$ are differentiable. If $f'(2)=18f(2)$, $\alpha'(2)=3\alpha(2)$, $\beta'(2)=-4\beta(2)$, and $\gamma'(2)=k\gamma(2)$, then what is the value of $k$?
• For the piecewise-defined function $$f(x)=\begin{cases} \dfrac{\sin((a+1)x)+\sin x}{x}, & x<0,\\[4pt] c, & x=0,\\[4pt] \dfrac{\sqrt{x+bx^2}-\sqrt{x}}{b x^{3/2}}, & x>0, \end{cases}$$ to be continuous at $x=0$, which values of $a$, $b$, and $c$ are required?
• Let $f(x)$ and $g(x)$ be continuously differentiable functions such that $f'(x)=g(x)$ and $f''(x)=-f(x)$. Define $h(x)=[f(x)]^2+[g(x)]^2$. If $h(0)=5$, what is the value of $h(10)$?
• For what value of $p$ can the function \[ f(x)=\begin{cases} \dfrac{(4^x-1)^3}{\sin x\,[\log|1+x^2/3|]^p}, & x\ne 0,\\[6pt] 12(\log 4)^3, & x=0, \end{cases} \] be made continuous at $x=0$?
• Using the mean value theorem, \(\dfrac{f(b)-f(a)}{b-a}=f'(c)\), take $a=0$, $b=\frac12$, and $f(x)=x(x-1)(x-2)$. What is the value of $c$?
• Consider the function $f:\mathbb{R}\to\mathbb{R}$ given by $f(x)=\max\{x,x^3\}$. At which points is $f(x)$ not differentiable?

⚔️ Practice JEE Main Maths free + battle 1v1 →