Recent questions tagged limits

Let $(v_n)$ be a sequence defined by $v_1=2$ and $v_{n+1}=\frac{1}{2}\left(v_n+\frac{8}{v_n}\right)$ for $n\geq1$. If $(v_n)$ converges, then $\lim_{n\to\infty}v_n$ is:$2...

Evaluate the limit $$\lim_{x\to0}\dfrac{e^{x^2}-\cos(2x)-3x\sin x}{x^4}$$$0$ $\frac{1}{3}$ $\frac{1}{2}$ $1$

The value of $\lim_{n\to\infty}\dfrac{1}{n}\left(\dfrac{n}{n+1}+\dfrac{n}{n+2}+\dfrac{n}{n+3}+\cdots+\dfrac{n}{2n}\right)$ is:$0$ $\ln 2$ $1$ $2\ln 2$

Let $L=\lim_{x\to 0}\left(\dfrac{e^{2x}+\sin x}{1+3x}\right)^{1/x^2}$. Find the value of $\log_e L$.

Which of the following statements is/are TRUE?If $\lim_{x\to a} f(x)$ exists and $\lim_{x\to a} g(x)$ exists, then $\lim_{x\to a} f(g(x))$ must exist. If $\lim_{x\to a} f...

Let $$f(x)=\frac{(x-3)(x^2+1)}{|x-3|} \text{ for } x\ne 3$$ If $f$ is extended to be continuous at $x=3$, what should be the value of $f(3)$?$10$ $-10$ $0$ No such value ...

Consider the piecewise function $$f(x)=\begin{cases}\dfrac{e^{Cx}-1+4x}{\sin(2x)},&x<0\\\\ \dfrac{(C^2-1)x^2+7x+3}{2x^2+\sqrt{x^4+1}},&x\ge 0\end{cases}$$where $C$ is a c...

A portion of the graph of an even function $f(x)$ is shown only on the interval $[0,6]$, although $f(x)$ is defined for all real numbers. The graph has a vertical asympto...

Evaluate the following limit and find the value of $a$:$$\lim_{x\to 0}\frac{\sqrt{1+4x+7x^2}-\sqrt[3]{1+6x+9x^2}+x-\ln(1+x)}{x^2}=\frac{a}{6}$$What is the value of $a$?

Determine the following limit:$$L=\lim_{x\to\infty}x^2\left(\sqrt{9x^2+6x+5}-3x-1-\frac{2}{3x}\right)$$Which of the following is the correct value of the limit?$0$ $\frac...

Suppose that a function $f$ is defined by $$f(x)=\begin{cases}\frac{x^2-2x-8}{x-4}+3, & x\lt 4 \\\\ 25, & x=4 \\\\ \frac{\sqrt{x+5}-3}{x-4}, & x\gt 4\end{cases}$$Let $M=\...

Evaluate the limit $\lim_{x\to-\infty}(\sqrt{9x^2+6x+5}+3x)$.$-1$ $0$ $1$ Limit does not exist

$$f(x)=\left(\frac{|x|}{2}-x\right) \times\left(x-\frac{|x|}{2}\right)$$$f$ has local minimum $f$ has local maximum $f^{\prime}$ is continuous at $x=0$ $f^{\prime}$ is di...

Let $f:[0,1] \longrightarrow \mathbb{R}$ and $g: \mathbb{R} \longrightarrow \mathbb{R}$ be continuous functions. Assume that $g$ is periodic with period $1.$ Show that\[\...

Prove or disprove the following statements:Suppose that $f(z)$ is a complex analytic function in the punctured unit disk $0<|z|<1$ such that $\lim _{n \rightarrow \infty}...

Show that the power series $\sum_{n=1}^{\infty} z^{n!}$ represents an analytic function $f(z)$ in the open unit disk $\Delta$ centred at $0.$ Show that $f(z)$ cannot be e...

$\displaystyle\lim_{x \to \tfrac{\pi}{2}} \left( \frac{\tan 2x}{x - \tfrac{\pi}{2}} \right)$ = ________.$0$$1$$2$Undefined

The value of the function $f(x)=\lim _{x \rightarrow 0} \frac{x^3+x^2}{2 x^3-7 x^2}$ is$0$ $-\frac{1}{7}$ $\frac{1}{7}$ $\infty$

The $\lim _{x \rightarrow 0} \frac{\sin \left[\frac{2}{3} x\right]}{x}$ is$\frac{2}{3}$ $1$ $\frac{3}{2}$ $\infty$

Evaluate the following limit:$$\lim _{x \rightarrow 2}\left[\frac{1}{x-2}-\frac{2(2 x-3)}{x^3-3 x^2+2 x}\right]$$$0$$1$$-\frac{1}{2}$$\frac{1}{2}$

Evaluate the following limit:$$ \lim _{x \rightarrow 0} \frac{e^{2 x}-2 x-\cos x}{x \sin x} $$

Let $x_{n}=\left(1-\frac{1}{\sqrt{n}}\right)^{n} \exp \left(n^{\frac{1}{4}}\right)$. Which of the following statements is correct?$\displaystyle \lim _{n \rightarrow \inf...

Simplify the expression: $ \lim _{x \rightarrow 4} \frac{x^2-16}{x-4} $10816