GATE CS Calculus: Limits, Continuity, Differentiability questions

Practise Calculus: Limits, Continuity, Differentiability questions for GATE CS. Explore official PYQs and original topic-wise quizzes, check answers and view available solutions. Free, no sign-in.

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SubjectTopicYearType

What is the value of

\displaystyle\lim_{n \to \infty}\left(1 - \frac{1}{n}\right)^{2n}

\displaystyle \lim_{x \to \infty}\frac{x-\sin x}{x+\cos x}

equals

Consider the following two statements about the function

f(x)=\left\vert x\right\vert

: P.

f(x)

is continuous for all real values of

x

. Q.

f(x)

is differentiable for all real values of

x

. Which of the following is TRUE?

Consider the function

y=|x|

in the interval

[-1, 1]

. In this interval, the function is

The formula used to compute an approximation for the second derivative of a function

f

at a point

x_0

Compute without using power series expansion

\displaystyle \lim_{x \to 0} \frac{\sin x}{x}.

The solution of differential equation

y''+3y'+2y=0

is of the form

If the cube roots of unity are

1, \omega

and

\omega^2

, then the roots of the following equation are

(x-1)^3 +8 =0

\displaystyle \lim_{x \to 0} \frac{x(e^x - 1) + 2(\cos x -1)}{x(1 - \cos x)}

is __________

The differential equation

\frac{d^2 y}{dx^2}+\frac{dy}{dx}+\sin y =0

is:

Fourier series of the periodic function (period 2π) defined by

f(x) = \begin{cases} 0, -p < x < 0\\x, 0 < x < p \end{cases} \text{ is }\\ \frac{\pi}{4} + \sum \left [ \frac{1}{\pi n^2} \left(\cos n\pi - 1 \right) \cos nx - \frac{1}{n} \cos n\pi \sin nx \right ]

But putting

x = \pi

, we get the sum of the series …

The equation

7x^{7}+14x^{6}+12x^{5}+3x^{4}+12x^{3}+10x^{2}+5x+7=0

has