Limits
Credit: www.math.cornell.edu
1. Good Questions in Limits
Question 1. Let $f$ be the function defined by $f(x)=\sin x+\cos x$ and let $g$ be the function defined by $g(u)=\sin u+\cos u$, for all real numbers $x$ and $u$. Then,
- (a) $f$ and $g$ are exactly the same functions
- (b) if $x$ and $u$ are different numbers, $f$ and $g$ are different functions
- (c) not enough information is given to determine if $f$ and $g$ are the same.
Question 2. TRUE or FALSE. If
- $f(x)=\displaystyle{\frac{x^2-4}{x-2}}$ and
- $g(x)=x+2$, then we can say the functions $f$ and $g$ are equal.
Question 3. Imagine that there is a rope around the equator of the earth. Add a 20 meter segment of rope to it. The new rope is held in a circular shape centered about the earth. Then the following can walk beneath the rope without touching it:
- (a) an amoeba
- (b) an ant
- (c) I (the student)
- (d) all of the above
Question 4. Given two infinite decimals $a=0. 3939393939…$ and $b=0.67766777666…$, their sum $a+b$
- (a) is not defined because the sum of a rational and irrational number is not defined.
- (b) is not a number because not all infinite decimals are real numbers.
- (c) can be defined precisely by using successively better approximations
- (d) is not a real number because the pattern may not be predictable indefinitely.
Question 5. TRUE or FALSE. As $x$ increases to $100$, $f(x)=1/x$ gets closer and closer to $0$, so the limit as $x$ goes to $100$ of $f(x)$ is $0$. Be prepared to justify your answer.
Question 6. TRUE or FALSE. $\displaystyle{\lim_{x\rightarrow a}f(x)=L}$ means that if $x_1$ is closer to $a$ than $x_2$ is, then $f(x_1)$ will be closer to $L$ than $f(x_2)$ is. Be prepared to justify your answer with an argument or counterexample.
Question 7. You’re trying to guess $\displaystyle{\lim_{x \rightarrow 0}f(x)}$. You plug in $x=0.1, 0.01, 0.001, \dots$ and get $f(x)=0$ for all these values. In fact, you’re told that for all $n=1, 2, \dots, f\left(\frac{1}{10^n}\right)=0$.
TRUE or FALSE: Since the sequence $0.1, 0.01, 0.001, \dots$ goes to $0$, we know $\displaystyle{\lim_{x \rightarrow 0} f(x)}=0$.
Question 8. Suppose you have an infinite sequence of closed intervals, each one contains the next, and suppose too that the width of the $n$th interval is less than $\frac{1}{n}$. If $a$ and $b$ are in each of these intervals,
- (a) $a$ and $b$ are very close but they don’t have to be equal
- (b) either $a$ or $b$ must be an endpoint of one of the intervals
- (c) $a=b$
Question 9. Consider the function $$f(x)=\left\{\begin{array}{ll} x^2 & \mbox{$x$ is rational, $x\neq 0$} \\ -x^2 & \mbox{$x$ is irrational} \\ \mbox{undefined} & x=0 \end{array}\right.$$ Then
- (a) there is no $a$ for which $\displaystyle{\lim_{x\rightarrow a}f(x)}$ exists
- (b) there may be some $a$ for which $\displaystyle{\lim_{x\rightarrow a}f(x)}$ exists, but it is impossible to say without more information
- (c) $\displaystyle{\lim_{x\rightarrow a}f(x)}$ exists only when $a=0$
- (d) $\displaystyle{\lim_{x\rightarrow a}f(x)}$ exists for infinitely many $a$
Question 10. The statement “Whether or not $\displaystyle{\lim_{x\rightarrow a}f(x)}$ exists, depends on how $f(a)$ is defined,” is true
- (a) sometimes
- (b) always
- (c) never
See more: Function Exercise
Question 11. If a function $f$ is not defined at $x=a$,
- (a) $\displaystyle{\lim_{x\rightarrow a} f(x)}$ cannot exist
- (b) $\displaystyle{\lim_{x\rightarrow a} f(x)}$ could be $0$
- (c) $\displaystyle{\lim_{x\rightarrow a} f(x)}$ must approach $\infty$
- (d) none of the above.
Question 12. If $\displaystyle{\lim_{x\rightarrow a} f(x)=0}$ and $\displaystyle{\lim_{x\rightarrow a} g(x)=0}$, then $\displaystyle{\lim_{x\rightarrow a} \frac{f(x)}{g(x)}}$
- (a) does not exist
- (b) must exist
- (c) not enough information
The following two problems to be used in a sequence:
Question 13. The reason that $\displaystyle{\lim_{x\rightarrow 0}\sin (1/x)}$ does not exist is:
- (a) because no matter how close $x$ gets to $0$, there are $x$’s near $0$ for which $\sin(1/x) =1$, and some for which $\sin (1/x)=-1$
- (b) because the function values oscillate around $0$
- (c) because $1/0$ is undefined
- (d) all of the above
Question 14. $\displaystyle{\lim_{x\rightarrow 0}x^2\sin (1/x)}$
- (a) does not exist because no matter how close $x$ gets to $0$, there are $x$’s near $0$ for which $\sin(1/x) =1$, and some for which $\sin (1/x)=-1$
- (b) does not exist because the function values oscillate around $0$
- (c) does not exist because $1/0$ is undefined
- (d) equals $0$
- (e) equals $1$
Question 15. Suppose you have two linear functions $f$ and $g$ shown below.
Then $\displaystyle{\lim_{x\rightarrow a}\frac{f(x)}{g(x)}}$ is
- (a) 2
- (b) does not exist
- (c) not enough information
- (d) 3
Question 16. TRUE or FALSE. Consider a function $f(x)$ with the property that $\displaystyle{\lim_{x\rightarrow a} f(x) =0}$. Now consider another function $g(x)$ also defined near $a$. Then $\displaystyle{\lim_{x\rightarrow a} [f(x)g(x)] = 0}$.
Question 17. TRUE or FALSE.
If $\displaystyle{\lim_{x\rightarrow a} f(x) =\infty}$ and $\displaystyle{\lim_{x\rightarrow a} g(x) =\infty}$, then $\displaystyle{\lim_{x\rightarrow a} [f(x)-g(x)] =0}$.
Question 18. Suppose you have two linear function $f$ and $g$ shown below.
Then $\displaystyle{\lim_{x\rightarrow \infty}\frac{f(x)}{g(x)}}$ is
- (a) 2
- (b) does not exist
- (c) not enough information
- (d) 3
Question 19. What is the maximum number of horizontal asymptotes that a function can have?
- (a) one
- (b) two
- (c) three
- (d) as many as we want
Question 20. TRUE or FALSE. A function can cross its horizontal asymptote.
2. Answer
Question 1. Let $f$ be the function defined by $f(x)=\sin x+\cos x$ and let $g$ be the function defined by $g(u)=\sin u+\cos u$, for all real numbers $x$ and $u$. Then,
- (a) $f$ and $g$ are exactly the same functions
- (b) if $x$ and $u$ are different numbers, $f$ and $g$ are different functions
- (c) not enough information is given to determine if $f$ and $g$ are the same.
Answer: (a). Both $f$ and $g$ are given by the same rule, and are defined on the same domain, hence they are the same function.
Question 2. TRUE or FALSE. If
- $f(x)=\displaystyle{\frac{x^2-4}{x-2}}$ and
- $g(x)=x+2$, then we can say the functions $f$ and $g$ are equal.
Answer: FALSE. Note that even if the two functions have the same rule, they are defined on different domains, i.e., $f$ is not defined at 2.
Question 3. Imagine that there is a rope around the equator of the earth. Add a 20 meter segment of rope to it. The new rope is held in a circular shape centered about the earth. Then the following can walk beneath the rope without touching it:
- (a) an amoeba
- (b) an ant
- (c) I (the student)
- (d) all of the above
Answer: (d). This question is quite difficult for students because it is very counter-intuitive. A little algebra needs to be done to see that as long as the student is not over $\frac{20}{2\pi}$ meters tall, she should be able to walk under the rope.
Students should know or be provided with the perimeter of a circle. There is no need to know the radius of the Earth at equator. The problem encourages using a mathematical model to check one’s intuition. Instructors should validate students’ intuition: the change in radius is very small relative to the radius, and this may lead to the erroneous conclusion that a human would not be able to walk underneath the rope; however, a human’s height is also very small relative to the radius.
Question 4. Given two infinite decimals $a=0. 3939393939…$ and $b= 0. 67766777666…$, their sum $a+b$
- (a) is not defined because the sum of a rational and irrational number is not defined.
- (b) is not a number because not all infinite decimals are real numbers.
- (c) can be defined precisely by using successively better approximations
- (d) is not a real number because the pattern may not be predictable indefinitely.
Answer: (c). Students may be unsure about real numbers as infinite decimals. Students know that all rational numbers have terminating or repeating decimal representations. They also know that there are irrational numbers, hence there are some numbers that are represented as infinite decimals. However, they may not know that every infinite decimal represents a number (although not uniquely in the case of repeating 9’s and repeating 0’s) -The phrase “can be defined precisely” may cause some to reject this as a solution. In discussing this question, instructors can introduce the idea that every infinite decimal is a number and the Archimedian Axiom can help us see how we can tell whether two numbers are the same.
Question 5. TRUE or FALSE. As $x$ increases to $100$, $f(x)=1/x$ gets closer and closer to $0$, so the limit as $x$ goes to $100$ of $f(x)$ is $0$. Be prepared to justify your answer.
Answer: FALSE. As $x$ increases to $100$, $f(x)=1/x$ gets closer and closer to $0$, gets closer and closer to $1/1000$, but not as close as to $1/100$.
The question points out the weakness of the statement “$f(x)$ gets closer to $L$ as $x\to a$, and therefore $\displaystyle{\lim_{x\rightarrow a}f(x)=L}$”.
Question 6. TRUE or FALSE. $\displaystyle{\lim_{x\rightarrow a}f(x)=L}$ means that if $x_1$ is closer to $a$ than $x_2$ is, then $f(x_1)$ will be closer to $L$ than $f(x_2)$ is. Be prepared to justify your answer with an argument or counterexample.
Answer: FALSE. Going to the limit is not monotonic! As a counterexample you can consider $$ f(x)=\left\{ \begin{array}{cl}
2x & x\ge 0 \\ -x & x<0 \end{array}\right. $$ Then $\displaystyle{\lim_{x\rightarrow 0}f(x)=0}$, and take $x_1=0.25$, $x_2=-0.35$.
Question 7. You’re trying to guess $\displaystyle{\lim_{x \rightarrow 0}f(x)}$. You plug in $x=0.1, 0.01, 0.001, \dots$ and get $f(x)=0$ for all these values. In fact, you’re told that for all $n=1, 2, \dots,f\left(\frac{1}{10^n}\right)=0$.
TRUE or FALSE: Since the sequence $0.1, 0.01, 0.001, \dots$ goes to $0$, we know $\displaystyle{\lim_{x \rightarrow 0} f(x)}=0$.
Answer: FALSE. The goal is to see whether the students understand that it’s not enough to check the limit for one particular sequence of numbers that goes to 0. The instructor may want to recall the function $\displaystyle{\sin (\frac{\pi}{x})}$ from Stewart, as $x$ goes to 0, in order to discuss the problem. Make sure to point out this problem as an example of the danger of using calculators to “find” limits.
Question 8. Suppose you have an infinite sequence of closed intervals, each one contains the next, and suppose too that the width of the $n$th interval is less than $\frac{1}{n}$. If $a$ and $b$ are in each of these intervals,
- (a) $a$ and $b$ are very close but they don’t have to be equal
- (b) either $a$ or $b$ must be an endpoint of one of the intervals
- (c) $a=b$
Answer: (c). If using this problem, the instructor should briefly talk about the Archimedian Axiom, and how intersection of nested closed intervals $I_n$ of respective lengths $\frac{1}{n}$, is a single point. Since both $a$ and $b$ are in each of these $I_n$, this single point of intersection is $a=b$. Students have a hard time understanding the Squeeze Theorem, so this might be a good place to start in attacking that problem.
Question 9. Consider the function $$f(x)=\left\{\begin{array}{ll} x^2 & \mbox{$x$ is rational, $x\neq 0$} \\ -x^2 & \mbox{$x$ is irrational} \\ \mbox{undefined} & x=0 \end{array}\right.$$ Then
- (a) there is no $a$ for which $\displaystyle{\lim_{x\rightarrow a}f(x)}$ exists
- (b) there may be some $a$ for which $\displaystyle{\lim_{x\rightarrow a}f(x)}$ exists, but it is impossible to say without more information
- (c) $\displaystyle{\lim_{x\rightarrow a}f(x)}$ exists only when $a=0$
- (d) $\displaystyle{\lim_{x\rightarrow a}f(x)}$ exists for infinitely many $a$
Answer: (c). Students should be encouraged to draw the graph and discuss.
Question 10. The statement “Whether or not $\displaystyle{\lim_{x\rightarrow a}f(x)}$ exists, depends on how $f(a)$ is defined,” is true
- (a) sometimes
- (b) always
- (c) never
Answer: (c). Use this problem to stress that $f(a)$ need not be defined in order for $\displaystyle{\lim_{x\rightarrow a}f(x)}$ to exist. Students have a difficult time asserting “never”. The problem provides an opportunity to discuss what a limit is.
Question 11. If a function $f$ is not defined at $x=a$,
- (a) $\displaystyle{\lim_{x\rightarrow a} f(x)}$ cannot exist
- (b) $\displaystyle{\lim_{x\rightarrow a} f(x)}$ could be $0$
- (c) $\displaystyle{\lim_{x\rightarrow a} f(x)}$ must approach $\infty$
- (d) none of the above.
Answer: (b). Answers $(a)$ and $(c)$ are very popular. $f(a)$ need not be defined in order for $\displaystyle{\lim_{x\rightarrow a}f(x)}$ to exist, and it does not have to approach $\infty$. However, the limit could be 0, for example consider $f(x)= 0$ for all $x \neq a$, and $f(a)$ not defined. The student has to note the difference between “cannot”, “could” and “must”.
Question 12. If $\displaystyle{\lim_{x\rightarrow a} f(x)=0}$ and $\displaystyle{\lim_{x\rightarrow a} g(x)=0}$, then $\displaystyle{\lim_{x\rightarrow a} \frac{f(x)}{g(x)}}$
- (a) does not exist
- (b) must exist
- (c) not enough information
Answer: (c). Point out that $\frac{0}{0}$ is not always equal to $1$. If this question is used after any of the previous two problems, more students will be able to answer correctly.
The following two problems to be used in a sequence:
Question 13. The reason that $\displaystyle{\lim_{x\rightarrow 0}\sin (1/x)}$ does not exist is:
- (a) because no matter how close $x$ gets to $0$, there are $x$’s near $0$ for which $\sin(1/x) =1$, and some for which $\sin (1/x)=-1$
- (b) because the function values oscillate around $0$
- (c) because $1/0$ is undefined
- (d) all of the above
Answer: (a). Illustrate why (b) and (c) are not the reason why the limit does not exist, by introducing the next problem.
Question 14. $\displaystyle{\lim_{x\rightarrow 0}x^2\sin (1/x)}$
- (a) does not exist because no matter how close $x$ gets to $0$, there are $x$’s near $0$ for which $\sin(1/x) =1$, and some for which $\sin (1/x)=-1$
- (b) does not exist because the function values oscillate around $0$
- (c) does not exist because $1/0$ is undefined
- (d) equals $0$
- (e) equals $1$
Answer: (d). As in the previous problem, the function oscillates and $1/0$ is undefined, however, this limit exists. This is also a nice application of The Squeeze Theorem: $$\displaystyle{\lim_{x\rightarrow 0} (-x^2)}\le \displaystyle{\lim_{x\rightarrow 0}x^2\sin (1/x)} \le \displaystyle{\lim_{x\rightarrow 0} x^2}$$ Therefore, the limit equals $0$.
Question 15. Suppose you have two linear functions $f$ and $g$ shown below.
Then $\displaystyle{\lim_{x\rightarrow a}\frac{f(x)}{g(x)}}$ is
- (a) 2
- (b) does not exist
- (c) not enough information
- (d) 3
Answer: (a). This problem requires a geometrical argument:
Solution 1: By similar triangles, $\frac{f(x)}{6}=\frac{ x-a}{0-a}=\frac{ g(x)}{3}$, and therefore $\frac {f(x)}{g(x)}=\frac {6}{3}=2$.
Solution 2: $$\displaystyle{\lim_{x\rightarrow a} \frac {f(x)}{g(x)}}=\displaystyle{\lim_{x\rightarrow a} \frac {\frac{f(x)}{-a}}{\frac{g(x)}{-a}}}=\displaystyle{\lim_{x\rightarrow a} \frac {\mbox{slope of } f}{\mbox{slope of } g}}=\frac{6}{3}=2$$ This problem is a nice preview of L’Hospital’s Rule.
Question 16. TRUE or FALSE. Consider a function $f(x)$ with the property that $\displaystyle{\lim_{x\rightarrow a} f(x) =0}$. Now consider another function $g(x)$ also defined near $a$. Then $\displaystyle{\lim_{x\rightarrow a} [f(x)g(x)] = 0}$.
Answer: FALSE. Students might justify a True answer by “zero times any number equals zero”. Point out that it is possible that $\displaystyle{\lim_{x\rightarrow a} g(x) =\infty}$. A quick counterexample can be $a=0$, $f(x)=x$ and $g(x)=1/x$.
Question 17. TRUE or FALSE.
If $\displaystyle{\lim_{x\rightarrow a} f(x) =\infty}$ and $\displaystyle{\lim_{x\rightarrow a} g(x) =\infty}$, then $\displaystyle{\lim_{x\rightarrow a} [f(x)-g(x)] =0}$.
Answer: FALSE. Students might be thinking that $\infty$ is a number, and therefore $\infty -\infty=0$. As a quick counterexample, consider $f(x)=x^2$ and $g(x)=x$.
Question 18. Suppose you have two linear function $f$ and $g$ shown below.
Then $\displaystyle{\lim_{x\rightarrow \infty}\frac{f(x)}{g(x)}}$ is
- (a) 2
- (b) does not exist
- (c) not enough information
- (d) 3
Answer: (a). Recall problem $6.$ in Section $2.3$. $\frac{f(x)}{6}=\frac{ x-a}{0-a}=\frac{ g(x)}{3}$, and therefore $\frac {f(x)}{g(x)}=\frac {6}{3}=2$.
Question 19. What is the maximum number of horizontal asymptotes that a function can have?
- (a) one
- (b) two
- (c) three
- (d) as many as we want
Answer: (b). Students must pay attention to the way horizontal asymptotes are defined. Point out that asymptotes are defined as we go to $\infty$ and to $-\infty$, even though a function may have asymptotic behavior at other points.
Question 20. TRUE or FALSE. A function can cross its horizontal asymptote.
Answer: TRUE. It is easy to sketch a function that crosses its horizontal asymptote. For example, consider $\frac{\sin x}{x}$.
