Recent questions in Calculus

Let $K$ be a cube of side $1$ in $\mathbb{R}^{3}$, with its centre at the origin, and its sides parallel to the co-ordinate axes. For $t \geq 0$, let $K_{t}$ be the set o...

Let $\lim _{n \rightarrow \infty} f(n)=\infty$ and $\lim _{n \rightarrow \infty} g(n)=\infty$. Then which of the following is necessarily $\text{TRUE.}$$\lim _{n \rightar...

Consider\[f(x)=\frac{(x \log x+x)^{5}(1+2 / x)^{x}}{(x+1 / x)^{5}(\log x+1 / \log x)^{6}}\]What can we say about $\lim _{x \rightarrow \infty} f(x)$ ?The function $f(x)$ ...

Consider real-valued continuous functions $f:[0,2] \rightarrow(-\infty, \infty)$ and let\[A=\int_{0}^{1}|f(x)| d x \quad \text { and } B=\int_{1}^{2}|f(x)| d x .\]Which o...

Let $a, b \in\{0,1\}$. Consider the following statements where $*$ is the $\text{AND}$ operator, $\oplus$ is $\text{EXCLUSIVE-OR,}$ and ${ }^{c}$ denotes the complement f...

Consider the positive integer sequence\[x_{n}=n^{50} e^{-(\log (n))^{3 / 2}}, \quad n=1,2,3, \ldots\]Which of the following statements is $\text{TRUE?}$For every $M>0$, t...

Suppose that $f(x)$ is a real valued continuous function such that $f(x) \rightarrow \infty$ as $x \rightarrow \infty$. Further, let\[a_{n}=\sum_{j=1}^{n} 1 / j\]and\[b_{...

Suppose $f(x)=c x^{-\alpha}$ for some $c>0$ and $\alpha>0$ such that $\int_{1}^{\infty} f(x) \mathrm{d} x=1$. Then, which of the following is possible?$\int_{1}^{\infty} ...

If $f(t)=\frac{\omega}{s^2+\omega^2}$, then the value of $\lim _{t \rightarrow \infty} f(t)$cannot be determinedis zerois unityis infinite

The derivative of the symmetric function drawn in given figure will look like

The integral $\displaystyle{}\int_0^\pi \sin ^3 \theta\; d \theta$ is given by$\frac{1}{2}$$\frac{2}{3}$$\frac{4}{3}$$\frac{8}{3}$

As $x$ is increased from $-\infty$ to $\infty$, the function $$ f(x)=\frac{e^x}{1+e^x} $$monotonically increasesmonotonically decreasesincreases to a maximum value and th...

The following plot shows a function $y$ which varies linearly with $x$. The value of the integral $I=\displaystyle{}\int_1^2 y d x$ is$1.0$$2.5$$4.0$$5.0$

For $|x| \ll 1$, $\operatorname{coth}(x)$ can be approximated as$x$$x^2$$\frac{1}{x}$$\frac{1}{x^2}$

$\displaystyle{}\lim _{\theta \rightarrow 0} \frac{\sin (\theta / 2)}{\theta}$ is$0.5$$1$$2$not defined

Which one of the following functions is strictly bounded?$\frac{1}{x^2}$$e^x$$x^2$$e^{-x^2}$

For the function $e^{-x}$, the linear approximation around $x=2$ is$(3-x) e^{-2}$$1-x$$[3+2 \sqrt{2}-\left(1+\sqrt{2}\right) x] e^{-2}$$e^{-2}$

Consider the function $f(x)=x^{2}-x-2$. The maximum value of $f(x)$ in the closed interval $[-4,4]$ is$18$$10$$-2.25$indeterminate

For real values of $x$, the minimum value of the function $f(x)=\exp (x)+\exp (-x)$ is$2$$1$$0.5$$0$

Which of the following functions would have only odd powers of $x$ in its Taylor series expansion about the point $x=0?$ $\sin \left(x^{3}\right)$$\sin \left(x^{2}\right)...