List of top Engineering Mathematics Questions on Calculus asked in GATE ES

Evaluate the limit:

\[ \lim_{x \to 0} \frac{x^2}{\sin x} \]

= _______.

Consider a function \[ y = f(x) \text{ which satisfies the following equation:} \] \[ \frac{d^2y}{dx^2} - \frac{dy}{dx} = 0. \] As \( x \to -\infty, y = 1 \), and at \( x = 0, y = 2 \). The value of \[ \frac{dy}{dx} \text{ at } x = 0 \text{ is } _______ \text{ (answer in integer).}

The ordinary differential equation \[ \frac{dy}{dx} = x^2 y \] has \( y \) as the dependent variable and \( x \) as the independent variable. Which of the following classification(s) is/are applicable to the equation?

A tangent is drawn on the curve of the function \( y = x^2 \) at the point \( (x, y) = (3, 9) \). The slope of the tangent is _______.