Limits Assignment.sagews

Limits Assignment

Question 0

Watch the lecture video here.

Did you watch the video? [Type yes or no.]

Question 1

Consider $\displaystyle\lim_{x\to-2}\frac{x^2-x-6}{x^3-6x^2+32}$.

Part a

Estimate the limit to two decimal places by zooming in on a graph.

#The limit is approximately:

Part b

Estimate the limit numerically from the left using at least seven values.

#The limit is approximately:

Part c

Estimate the limit numerically from the right using at least seven values.

#The limit is approximately:

Part d

Compute the limit using Sage's limit command. [Convert your answer to a decimal in order to compare it with the results above.]

Question 2

Consider $\displaystyle\lim_{x\to1}\frac{x^3-1}{\sqrt{x}-1}$.

Part a

Estimate the limit to two decimal places by zooming in on a graph.

#The limit is approximately:

Part b

Estimate the limit numerically from the left using at least seven values.

#The limit is approximately:

Part c

Estimate the limit numerically from the right using at least seven values.

#The limit is approximately:

Part d

Compute the limit using Sage's limit command.

Question 3

Consider $\displaystyle\lim_{x\to0}\frac{x}{|x|}$.

Note $|x|$=abs(x) in Sage.

Part a

Estimate the limit to two decimal places by zooming in on a graph.

#The limit is approximately:

Part b

Estimate the limit numerically from the left using at least seven values.

#The limit is approximately:

Part c

Estimate the limit numerically from the right using at least seven values.

#The limit is approximately:

Part d

Compute the limit using Sage's limit command.

Question 4

Let $\displaystyle f(x)=\frac{x^2-4}{x^2-1}$.

Part a

Use Sage's limit command to compute the right limit $\displaystyle\lim_{x\to1^+}f(x)$.

Part b

Use Sage's limit command to compute the left limit $\displaystyle\lim_{x\to1^-}f(x)$.

Question 5

Consider the function $\displaystyle f(x)=\frac{12x^4-9x^2+8}{3x^4+2x^3-4x}$.

Part a

Compute $\displaystyle\lim_{x\to-\infty}f(x)$.

Part b

Compute $\displaystyle\lim_{x\to\infty}f(x)$.

Part c

Graph $f(x)$ using xmin=-100, xmax=100, ymin=0, ymax=8. Your graph should have a horizontal asymptote that matches the answers from parts a and b.

Question 6

Let $f(x)=\sqrt{2x+4}$. Compute $\displaystyle\lim_{b\to a}\frac{f(b)-f(a)}{b-a}$.

This material was developed by Aaron Tresham at the University of Hawaii at Hilo and is
licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.