함수의 극한과 연속 전체 문제

Yes

Yes

Yes

(a) No; $\displaystyle \lim_{x\to -2}f(x) \neq f(-2)$ (b) Yes (c) No; $f(2)$ is not defined.

Yes; $\displaystyle \lim_{x\to 3\pi/2} \sin x +1 = 0$, and $\sin(3\pi/2)+1 = 0$.

(a) $\infty$ (b) $\infty$

(a) $-\infty$ (b) $\infty$ (c) Limit does not exist (d) $\infty$ (e) $\infty$ (f) $\infty$

(a) $1$ (b) $0$ (c) $1/2$ (d) $1/2$

(a) Limit does not exist (b) Limit does not exist

(a) Limit does not exist (b) Limit does not exist

(a) $10$ (b) $\infty$

The velocity is positive when the object is going in the increasing direction; it is going “up" on the graph when $t$ is between 0 and 2, and when $t$ is between $6$ and $7$. So, the velocity is positive when $t$ is in $(0,2) \cup (6,7)$.

(a) $\displaystyle \lim_{x \rightarrow -2} f(x)=1$: as $x$ gets very close to $-2$, $y$ gets very close to $1$. (b) $\displaystyle \lim_{x \rightarrow 0}f(x)=0$: as $x$ gets very close to $0$, $y$ also gets very close to $0$. (c) $\displaystyle \lim_{x \rightarrow 2}f(x)=2$: as $x$ gets very close to $2$, $y$ gets very close to $2$. We ignore the value of the function where $x$ is exactly $2$.

The limit does not exist. As $x$ approaches 0 from the left, $y$ approaches -1; as $x$ approaches 0 from the right, $y$ approaches 1. This tells us $\displaystyle\lim_{x \rightarrow 0^-} f(x)=-1$ and $\displaystyle\lim_{x \rightarrow 0^+} f(x)=1$, but neither of these are what the question asked. Since the limits from left and right do not agree, the limit does not exist. Put another way, there is no single number $y$ approaches as $x$ approaches 0, so the limit $\displaystyle\lim_{x \rightarrow 0} f(x)$ does not exist.

(a) $\displaystyle \lim_{x \rightarrow -1^{-}} f(x)=2$: as $x$ approaches $-1$ from the left, $y$ approaches 2. It doesn't matter that the function isn't defined at $x=-1$, and it doesn't matter what happens to the right of $x=-1$. (b) $\displaystyle \lim_{x \rightarrow -1^{+}} f(x)=-2$: as $x$ approaches $-1$ from the right, $y$ approaches -2. It doesn't matter that the function isn't defined at $-1$, and it doesn't matter what happens to the left of $-1$. (c) $\displaystyle \lim_{x \rightarrow -1} f(x) =$ DNE: since the limits from the left and right don't agree, the limit does not exist. (d) $\displaystyle \lim_{x \rightarrow -2^{+}} f(x) =0$: as $x$ approaches $-2$ from the right, $y$ approaches 0. It doesn't matter that the function isn't defined at 2, or to the left of 2. (e) $\displaystyle \lim_{x \rightarrow 2^{-}} f(x)=0$: as $x$ approaches $2$ from the left, $y$ approaches 0. It doesn't matter that the function isn't defined at 2, or to the right of 2.

We can begin by plotting the points that are easy to read off the diagram. $$\begin{array}{c|c|c} x&f(x)&\frac{1}{f(x)}\\ \hline -3&-3&\frac{-1}{3}\\ -2&0&UND\\ -1&3&\frac{1}{3}\\ 0&3&\frac{1}{3}\\ 1&\frac{3}{2}&\frac{2}{3}\\ 2&0&UND\\ 3&1&1 \end{array}$$ Note that $\frac{1}{f(x)}$ is undefined when $f(x) = 0$. So $\frac{1}{f(x)}$ is undefined at $x=-2$ and $x=2$. We shall look more closely at the behaviour of $\frac{1}{f(x)}$ for $x$ near $\pm 2$ shortly.

Plotting the above points, we get the following picture:

Since $f(x)$ is constant when $x$ is between -1 and 0, then also $\frac{1}{f(x)}$ is constant between -1 and 0, so we update our picture:

The big question that remains is the behaviour of $\frac{1}{f(x)}$ when $x$ is near -2 and 2. We can answer this question with limits. As $x$ approaches $-2$ from the left, $f(x)$ gets closer to zero, and is negative. So $\frac{1}{f(x)}$ will be negative, and will increase in magnitude without bound; that is, $\displaystyle\lim_{x \rightarrow -2^-}\dfrac{1}{f(x)}=-\infty$. Similarly, as $x$ approaches $-2$ from the right, $f(x)$ gets closer to zero, and is positive. So $\frac{1}{f(x)}$ will be positive, and will increase in magnitude without bound; that is, $\displaystyle\lim_{x \rightarrow -2^+}\dfrac{1}{f(x)}=\infty$. We add this behaviour to our graph:

Now, we consider the behaviour at $x=2$. Since $f(x)$ gets closer and closer to 0 AND is positive as $x$ approaches 2, we conclude $\displaystyle\lim_{x \rightarrow 2} \frac{1}{f(x)}=\infty$. Adding to our picture:

Now the only remaining blank space is between $x=0$ and $x=1$. Since $f(x)$ is a smooth curve that stays away from 0, we can draw some kind of smooth curve here, and call it good enough. (Later on we'll go into more details about drawing graphs. The purpose of this exercise was to utilize what we've learned about limits.)

We can start by examining points.

$$\begin{array}{c|c|c|c}x & f(x) & g(x) & \dfrac{f(x)}{g(x)} \\ \hline -3 & -3 & -1.5 & 2 \\ \hline -2 & 0 & 0 & \text{UND} \\ \hline -1 & 3 & 1.5 & 2 \\ \hline -0 & 3 & 1.5 & 2 \\ \hline 1 & 1.5 & .75 & 2 \\ \hline 2 & 0 & 0 & \text{UND} \\ \hline 3 & 1 & .5 & 2\end{array}$$

We cannot divide by zero, so $\dfrac{f(x)}{g(x)}$ is not defined when $x=\pm2$. But for every other value of $x$ that we plotted, $f(x)$ is twice as large as $g(x)$, $\dfrac{f(x)}{g(x)}=2$. With this in mind, we see that the graph of $f(x)$ is exactly the graph of $2g(x)$.

This gives us the graph below.

Remark: $f(2)=g(2)=0$, so $\dfrac{f(2)}{g(2)}$ does not exist, but $\displaystyle\lim_{x \rightarrow 2}\dfrac{f(x)}{g(x)}=2$. Although we are trying to “divide by zero" at $x=\pm 2$, it would be a mistake here to interpret this as a vertical asymptote.

Many answers are possible; the tangent function behaves like this.

One example is $f(x) = \left\{ \begin{array}{ll} 0&\text{when }0 \leq x \leq 1\\ 2&\text{when }1<x \leq 2 \end{array}\right.$. The IVT only guarantees $f(c)=1$ for some $c$ in $[0,2]$ when $f$ is continuous over $[0,2]$. If $f$ is not continuous, the IVT says nothing.

(a) One possible graph follows. (b) One possible graph follows.