Limits

For $x>0, \lim _{x \rightarrow 0}\left((\sin x)^{1 / x}+\left(\frac{1}{x}\right)^{\sin x}\right)$ is

Paragraph: Read the following passage and answer the questions. For every function $f(x)$ which is twice differentiable, these will be good approximation of $\int_a^b f(x) d x…

Let $f(x)$ be differentiable on the interval $(0, \infty)$ such that $f(1)=1$, and $\lim _{t \rightarrow x} \frac{t^2 f(x)-x^2 f(t)}{t-x}=1$ for each $x>0$. Then, $f(x)$ is

If $\lim _{x \rightarrow 0}\left[1+x \log \left(1+b^2\right)\right]^{1 / x}=2 b \sin ^2 \theta b>0$ and $\theta \in(-\pi, \pi]$, then the value of $\theta$ is

Let $f(x)=limn\to∞⁡(\frac{n^{n}(x+n)(x+\frac{n}{2})\ldots..(x+\frac{n}{n})}{n!(x^{2}+n^{2})(x^{2}+\frac{n^{2}}{4})\ldots..(x^{2}+\frac{n^{2}}{n^{2}})})^{\frac{x}{n}}$ , for all $x>0$ . Then

The value of $\lim _{x \rightarrow 0} \frac{1}{x^3} \int_0^x \frac{t \log (1+t)}{t^4+4} d t$ is

Let $F:R\to R$ be a function. We say that $f$ has: PROPERTY $1$ if $limh\to0⁡\frac{f(h)-f(0)}{\sqrt{|h|}}$ exists and is finite, and PROPERTY $2$ if $limh\to0⁡\frac{f(h)-f(0)}{h^{2}}$ exists and is finite. Then which of…

For $a\in R,|a|>1,$ let $limn\to∞⁡(\frac{1+\sqrt[3]{2}+\ldots+\sqrt[3]{n}}{n^{7/3}(\frac{1}{(an+1)^{2}}+\frac{1}{(an+2)^{2}}+\ldots+\frac{1}{(an+n)^{2}})})=54.$ Then the possible value(s) of $a$ is/are:

Let $\alpha(a)$ and $\beta(a)$ be the roots of the equation $(\sqrt[3]{1+a}-1) x^{2}+(\sqrt{1+a}-1) x+(\sqrt[6]{1+a}-1)=0$ where $a$ $>-1$. Then $\lim _{a \rightarrow 0^{+}} \alpha(a)$ and $\lim _{x \rightarrow 0^{+}}…

Let $x_0$ be the real number such that $e^{x_0}+x_0=0$. For a given real number $\alpha$, define $g(x)=\frac{3 x e^x+3 x-\alpha e^x-\alpha x}{3\left(e^x+1\right)}$ for all real numbers $x$. Then which one of the…

Let $f:R\to(0,∞)$ and $g:R\to R$ be twice differentiable functions such that $f^{''}$ and $g^{''}$ are continuous functions on $R$ . Suppose $f^{'}(2)=g(2)=0,f^{''}(2)\neq0$ and…

Let $f:(0,1)\toℝ$ be the function defined as $f(x)=\sqrt{n}$ if $x\in[\frac{1}{n+1},\frac{1}{n})$ where $n\inℕ$. Let $g:(0,1)\toℝ$ be a function such that $\int_{x^{2}}^{x}\sqrt{\frac{1-t}{t}}dt<g(x)<2\sqrt{x}$ for all…