Practice (32)

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Compute $$\lim_{x\to 4}\frac{3-\sqrt{x+5}}{x-4}$$

Is the $y=\frac{1}{x}$ a continuous function?

Show that $$\lim_{x\to 0}\ \frac{x}{\sin{x}}=1$$

Show the following sequence is convergent:

$$\frac{1}{1^2},\ \frac{1}{2^2},\ \frac{1}{3^2},\ \cdots,\ \frac{1}{n^2},\ \cdots$$

Show that the limit of $f(n)=\left(1+\frac{1}{n}\right)^n$ exits when $n$ becomes infinitely large.

Show that $$\lim_{x\to 0}\frac{e^x-1}{x}=1$$

Find the value of 


Determine if the following infinite series is convergent or divergent:

$$\sum_{n=2}^{\infty}\frac{1}{(\ln n)^{\ln \ln n}}$$

Compute $$\lim_{x\to 0}\frac{\int_0^x\sin(xt)^2dt}{x^5}$$

Find the value of $$\displaystyle\lim_{n\to\infty}\sum_{k=1}^{n}\frac{n+k}{n^2 + k}$$

Evaluate $\displaystyle\lim_{n\to\infty}S_n$ where

$$S_n = 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots + (-1)^{n-1}\frac{1}{n}$$

Determine the values of $\alpha$ and $\beta$ such that 


For $n=1, 2,\dots$, let $x_n=\displaystyle\sum_{k=n+1}^{9n}\frac{k}{9n^2 + k^2}$. Find the value of $\displaystyle\lim_{n\to\infty}x_n$.

Show that $1-\cos{x} < x^2$ holds for all $x > 0$.

It is well-known that the solution to the Fibonacci sequence is


Show that


Compute $$\lim_{n\to\infty}\left(\sqrt{n+1}-\sqrt{n}\right)$$

Show that $$\lim_{n\to\infty}\int_0^1 x^n(1-x)^n dx = 0$$

Show the following result without explicitly performing the integration: $$\lim_{n\to\infty}\int_0^1(1-x^2)^ndx = 0$$

Show the following result without explicitly performing the integration $$\lim_{n\to\infty}\int_0^{\frac{\pi}{2}}\sin^n{x}dx$$

Without explicitly evaluating the integral, show that

$$\lim_{n\to\infty}\int_1^2\ln^n{x}dx =0\quad\text{and}\quad\lim_{n\to\infty}\int_2^3\ln^n{x}dx = \infty$$

Consider the parabola $y=ax^2 + 2019x + 2019$. There exists exactly one circle which is centered on the $x$-axis and is tangent to the parabola at exactly two points. It turns out that one of these tangent points is $(0, 2019)$. Find $a$.

What is the smallest natural number $n$ for which the following limit exists?

$$\lim_{x\to 0}\frac{\sin^nx}{\cos^2x(1-\cos{x})^3}$$


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

Find all the inflection points of $$\left\{\begin{array}{rl} x &=\cot {t} \\ y&=\frac{\cos(2t)}{\sin{t}} \end{array}\right., t\in(0, \pi)$$

Compute $$\lim_{x\to 0}\frac{(1-\cos{x})^2}{x^2-x^2\cos^2{x}}$$