Математический анализ 1 — МИЭФ, 2022 midterm

On the first grid, sketch a graph of a function $f$ such that:

(a) $f(x)$ is odd.

(b) $y=f(x)$ has asymptote $y=x-1$ as $x\to+\infty$.

(c) $y=f(x)$ has only two discontinuities, both of jump type.

(d) $\lim_{x\to1+}f(x)=1$ and $\lim_{x\to-1+}f(x)=-3$.

(e) $f(0)=f(1)$.

Then on the second grid sketch

$$y=|f(|x|)-2|,$$

and label discontinuities.

Consider

$$f(x)=\begin{cases}\dfrac{\sqrt{1-\sin(\alpha x^2)}-\cos x}{x\arctan x},&x<0,\\\beta,&0\le x\le8,\\\dfrac{7-\sqrt{6x+1}}{x^2-13x+40},&x>8.\end{cases}$$

(a) Find $\lim_{x\to8+}f(x)$.

(b) Find $\lim_{x\to0-}f(x)$. Use the standard equivalences for $\arctan u$, $\sin u$, $\sqrt{1+u}-1$, and $1-\\cos u$ as $u\to0$.

(c) What values of $\alpha$ and $\beta$, if any, make $f$ continuous on $\mathbb R$? Explain.

(d) Find $\lim_{x\to-\infty}f(x)$. Justify.

Consider

$$f(x)=\frac{\ln|\ln x|}{x-e}.$$

(a) Find the domain $D$.

(b) Give all vertical asymptotes of $y=f(x)$, if any. Explain.

(c) Does $y=f(x)$ have a horizontal asymptote? Justify.

(d) Find $p\notin D$ such that $f$ is defined on both sides of $p$ and $\lim_{x\to p-}f(x)=\lim_{x\to p+}f(x)=L$. Use $\ln(1+u)\sim u$ to find $L$.

Consider the sequence in which each $1/n$ appears $n$ times:

$$1,\frac12,\frac12,\frac13,\frac13,\frac13,\frac14,\ldots$$

(a) Find $a_{2021}$.

(b) Find $\lim_{n\to\infty}2021^{a_n}$.

(c) Find $\lim_{n\to\infty}a_n\sqrt n$.

The graphs of functions $f$ (solid line) and $g$ (dashed line) are shown. At $x=p$, function $f$ has a removable discontinuity, while $g$ has a jump discontinuity. Moreover,

$$f(p)=g(p)=p.$$

Which statement is false?

Нужно проверить: "The figure is schematic; exact coordinates are not relevant."

1. $\lim_{x\to p}f(x)g(x)$ exists.
2. $\lim_{x\to p}(f(x)-g(x))$ exists.
3. $\lim_{x\to p}(f(x)+g(x))$ does not exist.
4. $\lim_{x\to0}\frac{f(x)}{g(x)}$ does not exist.

Find

$$\lim_{x\to-\infty}\frac{(3x-1)(2-4x)}{|1+3x-6x^2|}.$$

1. $-2$.
2. $-1$.
3. $1$.
4. $2$.

The domain of

$$g(x)=\sqrt{\frac1{|x+1|}-1}$$

is:

1. $(-\infty,-1)\cup(-1,0]$.
2. $[-2,0]$.
3. $[-2,-1)\cup(-1,0]$.
4. $[-2,+\infty)$.

The smallest period of

$$f(x)=\sin(12x)+\cos(18x)$$

is:

1. $\frac\pi{18}$.
2. $\frac\pi9$.
3. $\frac\pi6$.
4. $\frac\pi3$.

Functions $f$ and $g$ are defined for all $x$. Function $f$ is odd, and $g$ is even. Which functions must be odd?

I. $f(g(x))$.

II. $g(f(x))$.

III. $f(x)g(x)$.

1. I only.
2. II only.
3. III only.
4. I and III only.

Find

$$\lim_{x\to+\infty}\left(\frac{x+2310}{x-2310}\right)^{x+2310}.$$

1. $e^{1155}$.
2. $e^{2310}$.
3. $e^{4620}$.
4. $+\infty$.

Find

$$\lim_{x\to0}\frac{5x^3-2x^5}{3x^5-20x^3}.$$

1. $-\frac14$.
2. $-\frac23$.
3. $-\frac1{10}$.
4. $\frac53$.

The line $y=23$ is a horizontal asymptote for which function?

1. $y=\frac{\sin(46x)}{2x}$.
2. $y=\frac1{x-23}$.
3. $y=\frac{23x}{1-x}$.
4. $y=\frac{46x^2-x}{1+2x^2}$.

Which graph could represent

$$y=f(|2x-1|-1),\qquad f(x)=\frac1x?$$

Нужно проверить: "Exact candidate shapes should be checked in the source image."

1. a.
2. b.
3. c.
4. d.

Compute

$$\lim_{n\to\infty}\frac{24n^3+18n^2-14n+1}{363636n-242424n^2+5-6n^3}.$$

1. $-\frac1{10101}$.
2. $\frac2{30303}$.
3. $3$.
4. $-4$.

Find

$$\lim_{x\to2}\frac{\sqrt{6-x}-2}{\sqrt{3-x}-1}.$$

1. $\frac14$.
2. $\frac12$.
3. $2$.
4. $\infty$.

Which value of $\sigma$ makes $\phi$ continuous on $(-\infty,\infty)$?

$$\phi(x)= \begin{cases} \sigma x^2+5x, & x<2,\\[6pt] \dfrac{22}{\sigma-x}, & x\ge2. \end{cases}$$

1. $3$.
2. $1$.
3. $-1$.
4. $-3$.
5. None of them.

Which sequences are convergent?

I. $\left\{\frac{\ln n}{\ln(2n)}\right\}$.

II. $\left\{\frac{(\ln n)^2}{n}\right\}$.

III. $\left\{\frac{\ln n}{n^n}\right\}$.

1. II and III only.
2. I and III only.
3. I and II only.
4. I, II and III.

If $9^x-8^x$ is equivalent to $cx$ as $x\to0$, then $c=$

1. $3\ln2-2\ln3$.
2. $2\ln3-3\ln2$.
3. $\ln(3/2)$.
4. $\ln(2/3)$.

Find

$$\lim_{x\to-5}\frac{6-\sqrt{41+x}}{x+5}.$$

1. $-1$.
2. $-\frac16$.
3. $-\frac1{12}$.
4. The limit does not exist.

If $f(x)$ is continuous for all real numbers, which statements are true?

I. $f\left(\lim_{x\to x_0}x\right)=\lim_{x\to x_0}f(x)$.

II. $\lim_{n\to\infty}f(x_n)=f(L)$ for any sequence $\{x_n\}$ with $\lim x_n=L$.

III. If $f(a)>0$, then there exists $\delta>0$ such that $f(x)>0$ for $x\in(a-\delta,a+\delta)$.

1. I, II and III.
2. I only.
3. I and II only.
4. I and III only.

If $f$ is continuous for all $x$ such that $|x-23|\le10$, which statements must be true?

I. If $f(13)f(33)<0$, then there exists $\xi\in(13,33)$ such that $f(\xi)=0$.

II. There exists $\zeta\in[13,33]$ such that $f(\zeta)=\max_{13\le x\le33}f(x)$.

III. $f$ attains all values between $f(13)$ and $f(33)$.

1. I and II only.
2. II and III only.
3. I and III only.
4. I, II and III.

A sequence starts with $a_1=1$ and satisfies

$$a_n=\frac{a_{n-1}}2+\frac1{a_{n-1}}$$

for $n>1$. Given that the sequence converges to $L$, which is true?

1. $L=\frac1{\sqrt2}$.
2. $L=1$.
3. $L=\sqrt2$.
4. $L=2$.

Find

$$\lim_{x\to+\infty}\frac{\sqrt{x}+2\sqrt[3]{x}+5\sqrt[6]{x}}{\sqrt{4x}+\sqrt[3]{1+3x}}.$$

1. $\frac12$.
2. $\frac13$.
3. $\frac23$.
4. $1$.

The sequences $\{x_n\}$, $\{y_n\}$, and $\{z_n\}$ satisfy $x_n<y_n<z_n$. Which statements are false?

I. If $\{x_n\}$ and $\{z_n\}$ are convergent, then $\{y_n\}$ is also convergent.

II. If $\lim z_n=\lim x_n$, then $\lim y_n$ exists.

III. If all three sequences converge, then $\lim x_n<\lim y_n<\lim z_n$.

1. I only.
2. I and II only.
3. III only.
4. I and III only.

Function $f$ is defined for $x\ne\pm1$. The graph $y=f(x)$ has two vertical asymptotes $x=-1$ and $x=1$ and two horizontal asymptotes $y=-1$ and $y=1$.

Which limit exists?

Нужно проверить: "The source image should be checked for the direction of divergence at each asymptote."

1. $\lim_{x\to-1}\arctan f(x)$.
2. $\lim_{x\to1}\arctan f(x)$.
3. $\lim_{x\to-1}\sin f(x)$.
4. $\lim_{x\to1}\sin f(x)$.

Find the asymptote for

$$f(x)=\sqrt{121x^2+33x}$$

as $x\to-\infty$.

1. $2y=22x+3$.
2. $2y+22x=-3$.
3. $2y=22x-3$.
4. $y=-11x-3$.

Function $f(x)$ is defined for all values of $x$ and

$$\lim_{x\to0}xf(x)=1.$$

Which statement can be true?

1. $\lim_{x\to0-}f(x)=10$ and $\lim_{x\to0+}f(x)=23$.
2. $f$ is continuous at $x=0$.
3. $\lim_{x\to0-}f(x)=+\infty$ and $\lim_{x\to0+}f(x)=-\infty$.
4. $\lim_{x\to0-}f(x)=-\infty$ and $\lim_{x\to0+}f(x)=+\infty$.

Find

$$\lim_{x\to2}\left(\frac{x^2-2x-3}{2x^2-3x+1}\right)^{x^2+1}.$$

1. $0$.
2. $-1$.
3. $\infty$.
4. $\frac1e$.
5. The expression is not defined in a neighbourhood of $x=2$.