Математический анализ 1 — МИЭФ, 2021 midterm

Consider

$$f(x)=\frac{|x-1|}{1-x^2}.$$

(a) Find the domain of $f$.

(b) Find and classify all discontinuities of $f$. Give equations of all vertical asymptotes, if any.

(c) Find the range of $f$. Justify.

(d) Sketch $y=|2f(-x)|$ and label the discontinuities.

Let $f(x)=P(x)/Q(x)$, where $P$ and $Q$ are quadratic polynomials.

(a) Find a possible $Q(x)$ if $f$ has a removable discontinuity at $x=1$ and vertical asymptote $x=-2$.

(b) Using (a), find $f(x)$ if $f(2)=0$ and $\lim_{x\to1}f(x)=-3$. Explain every step.

(c) Using (b), find $\lim_{x\to-\infty}f(x)$, $\lim_{x\to0}f(x)$, and $\lim_{x\to+\infty}f(x)$.

Consider

$$f(x)=\begin{cases}\left(\frac{x-1}{x}\right)^{-x},&x<0,\\\sqrt{Ax^2+Bx+C},&x\ge0.\end{cases}$$

(a) Find $\lim_{x\to0-}f(x)$.

(b) Find an asymptote of $y=f(x)$ as $x\to-\infty$, or explain why none exists.

(c) Find $A$, $B$, and $C$ if $f$ is continuous at $0$ and has slant asymptote $y=x/2-2/3$ as $x\to+\infty$.

Sierpiński's construction starts with a $1\times1$ square. At each step, every remaining square is divided into $9$ smaller squares and the middle square is removed.

(a) Let $a_n$ be the area of an individual square after step $n$. Find $a_1,a_2,a_3$ and $a_n$.

(b) Let $s_n$ be the number of squares after step $n$. Given $s_1=1$, $s_2=8$, find $s_3,s_4$ and $s_n$.

(c) Let $c_n$ be the total area after step $n$. Find $c_n$ and $\lim c_n$.

(d) How does the limiting area change if at each step the inscribed circles in the middle squares are removed instead of the whole middle squares?

Function $f$ is defined for all $x\in\mathbb R$. It is continuous everywhere except at $x=-1$ and $x=1$. The figure shows a portion of the graph $y=f(x)$.

Which of the following limits does not exist?

Нужно проверить: "Coordinates are approximate from the source graph."

1. $\lim_{x\to\pi}f(\cos x)$.
2. $\lim_{x\to\pi/6}f(2\sin x)$.
3. $\lim_{x\to\pi/2}f(2\cos x)$.
4. $\lim_{x\to\pi/3}f(\sin x)$.

Find

$$\lim_{x\to0}\frac{\ln(\cos2x)}{\arctan(\sqrt[3]{1-x^2}-1)}.$$

1. $\frac32$.
2. $-2$.
3. $-\frac23$.
4. $6$.

Find the asymptote as $x\to-\infty$ for

$$g(x)=\frac{2x^2+5x+e^x}{\sqrt{4x^2+7}}.$$

1. $2y+2x=-5$.
2. $2y-2x=7$.
3. $2y-2x=5$.
4. $2y+2x=-7$.

Which expression is not an indeterminate form?

1. $\infty-\infty$.
2. $1^\infty$.
3. $\frac\infty\infty$.
4. $\frac0\infty$.

The terms of a decreasing sequence $\{a_n\}$ lie inside the interval

$$(10^{-24},1)$$

for all $n>2020$.

Which statements must be true?

I. $\{a_n\}$ is bounded.

II. $\{a_n\}$ is convergent.

III. $\{a_n\}$ is infinitesimally small.

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

Which statement is true about

$$\phi(x)=\frac1{\sqrt{x^4-8x}}?$$

1. The range of $\phi$ is $(0,+\infty)$.
2. The domain of $\phi$ is $(-\infty,0]\cup[2,+\infty)$.
3. The graph $y=\phi(x)$ has two vertical asymptotes $x=0$ and $x=-2$.
4. Function $\phi$ is decreasing.

Find $a+b$ given that

$$f(x)= \begin{cases} \log_3\left(\frac{2-x}{1-x}\right), & |x|<1,\\ x^2+ax+b, & |x|\ge1 \end{cases}$$

is continuous.

1. $a+b=-\frac12$.
2. $a+b=-1+\frac1{\ln3}$.
3. $a+b=1$.
4. $a+b=\frac{\ln3}{2}$.

Compute

$$\lim_{n\to\infty}\frac{n^2}{n+(n+1)+(n+2)+\cdots+(2n-1)+2n}.$$

1. $\frac12$.
2. $0$.
3. $\infty$.
4. $\frac23$.

Which graph could represent

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

Нужно проверить: "The source only gives small qualitative panels."

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

Find $k$ such that

$$\ln\left(\frac{2^x+3^x-5^x}{x}\right)\sim kx \qquad\text{as }x\to0.$$

1. $1$.
2. $\ln\left(\frac65\right)$.
3. $0$.
4. No such $k$ exists.

Find

$$\lim_{x\to2}\left(\frac{x^2-3x+4}{x^2+2x-6}\right)^{2/(x-2)}.$$

1. $e^{-10}$.
2. $e^{-5}$.
3. $e^{-4/5}$.
4. $e^2$.

Function $f(x)$ is continuous for all real numbers. Which statements are true?

I. $\lim_{x\to2020}f(x)=f(2020)$.

II. $\lim_{n\to\infty}f(x_n)=f(L)$ for every infinite sequence $\{x_n\}$ such that $\lim_{n\to\infty}x_n=L$.

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

The terms of infinite sequences $\{x_n\}$ and $\{y_n\}$ satisfy $0<x_n<y_n$ for all $n$.

Which statements are true?

I. If $\{x_n\}$ and $\{y_n\}$ are both convergent, then $\left\{\frac{x_n}{y_n}\right\}$ is also convergent.

II. If $\{y_n\}$ is infinitesimally small, then $\{x_n\}$ is also infinitesimally small.

III. If $\{x_n\}$ is infinitely large, then $\{y_n\}$ is divergent.

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

Which functions have a jump discontinuity?

$$u(x)=\frac{x^2-1}{|x-1|}, \qquad v(x)=\cos\left(\frac{x^2-1}{|x-1|}\right), \qquad w(x)=\ln\left(2+\frac{x^2-1}{|x-1|}\right).$$

1. $u$ only.
2. $u$ and $w$.
3. $u$, $v$, and $w$.
4. None of them.

The sign function is

$$\operatorname{sgn}x= \begin{cases} -1, & x<0,\\ 0, & x=0,\\ 1, & x>0. \end{cases}$$

Which limit does not exist?

1. $\lim_{x\to0+}\operatorname{sgn}(x)$.
2. $\lim_{x\to1-}\operatorname{sgn}(x-1)$.
3. $\lim_{x\to2}\operatorname{sgn}(x-2)$.
4. $\lim_{x\to3}\operatorname{sgn}^2(x-3)$.

Which infinite sequences are convergent?

I. $\left\{\frac{\sqrt[n]{n}}{2.4^n}\right\}$.

II. $\left\{\frac{\sqrt[n]{n}}{\sqrt[n]{n^3+2020}}\right\}$.

III. $\left\{\frac{\sqrt[n]{n}}{\ln(n^n)}\right\}$.

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

An even function $f(x)$ has a Type II discontinuity at $x=0$. Which statements could be true?

I. $\frac1{f(x)}$ has a removable discontinuity at $x=0$.

II. $\frac1{f(x)}$ has a jump discontinuity at $x=0$.

III. $\frac1{f(x)}$ has a Type II discontinuity at $x=0$.

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

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.

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. $1$.

Let $f$ be a strictly increasing continuous function defined for all $x$.

Which statements must be true?

I. $f^2(x)$ is increasing.

II. $f^3(x)$ is increasing.

III. The inverse function $f^{-1}(x)$ is defined and increasing on the range of $f$.

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

Function $f$ is defined for all $x\in\mathbb R$ and is discontinuous at $x=-1$ and $x=1$. The figure shows a portion of $y=f(x)$.

Which expression is the biggest?

Нужно проверить: "Coordinates are approximate from the printed graph."

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

Find

$$\lim_{x\to+\infty}(3^x+3^{2x})^{1/x}.$$

1. $3e^3$.
2. $9$.
3. $12$.
4. $3e$.

Which sequence is bounded?

1. $\sqrt n\cos\left(\frac{\pi n}{4}\right)$.
2. $1-n+2n-3n^3$.
3. $2^{1-n+2n-3n^3}$.
4. $n^2\sin\left(\frac1n\right)$.

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 limit is not defined.