Recent questions tagged limits

Suppose $f : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function such that $f(x) = \frac{2 – \sqrt{x+4}}{\sin 2x}$ for all $x \neq 0.$ Then the value of $f(0)$ is...

Let $\{ f_{n}\}$ be a sequence of functions defined as follows:$$f_{n}(x) = x^{n} \cos (2 \pi nx), \; x \in [ – 1, 1].$$Then $\lim_{x \rightarrow 0} f_{n} (x)$ exists if ...

Given a real number $\alpha \in (0, 1),$ define a sequence $\{ x_{n}\}_{n \geq 0}$ by the following recurrence relation:$$x_{n+1} = \alpha x_{n} + (1 – \alpha) x_{n – 1},...

Consider the function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ defined by $f(x, y)=x^{2}(y-1)$. For $\vec{u}=\left(\frac{1}{2}, \frac{1}{2}\right)$ and $\vec{v}=(3,4)$,...

Let $f : \mathbb{R}\rightarrow \mathbb{R}$ be a continuously differentiable function and $f(1) = 4$. Then the value of$$\lim_{x\rightarrow 1}\int_{4}^{f(x)}\frac{2t}{x - ...

Evaluate the question of the following limits. $\lim_{x\rightarrow \infty} \frac{2x^{3}+3x-5}{5x^{3}+1}$

For each positive integer $n$, let$$s_n=\frac{1}{\sqrt{4n^2-1^2}}+\frac{1}{\sqrt{4n^2-2^2}}+\dots+\frac{1}{\sqrt{4n^2-n^2}}$$Then the $\displaystyle \lim_{n\rightarrow \i...

The value of $$\displaystyle\lim_{n\rightarrow\infty}\prod_{k=2}^{n}\left(1-\frac{1}{k^2}\right)$$is$1/2$$1$$1/4$$0$

Let $A$ be the set of all real numbers $\lambda \in [0,1]$ such that$$\displaystyle\lim_{p\rightarrow 0}\frac{\log(\lambda2^p+(1-\lambda)3^p)}{p}=\lambda \log2+(1-\lambda...