GATE Questions & Answers of Limit, Continuity and differentiability

What is the Weightage of Limit, Continuity and differentiability in GATE Exam?

Total 5 Questions have been asked from Limit, Continuity and differentiability topic of Calculus subject in previous GATE papers. Average marks 1.40.

$\underset{x\to \infty }{\mathrm{lim}}\left(\frac{x+\mathrm{sin}x}{x}\right)$ equals to

With reference to the conventional Cartesian (x, y) coordinate system, the vertices of a triangle have the following coordinates: (x1, y1) = (1, 0); (x2, y2) = (2, 2); and (x3, y3) = (4, 3). The area of the triangle is equal to

The expression $\underset{a\to 0}{\mathrm{lim}}\frac{{x}^{a}-1}{a}$ is equal to

What should be the value of λ such that the function defined below is continuous at x=$\frac{\mathrm{\pi }}{2}$
$f\left(x\right)=\left\{\begin{array}{ll}\frac{\lambda \mathrm{cos}x}{\frac{\pi }{2}-x}& ifx\ne \pi }{2}\\ 1& ifx=\pi }{2}\end{array}\right\$
The $\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{sin}\left[{}_{3}{}^{2}x\right]}{x}$ is