1.
Evaluate the limit \(\displaystyle\lim_{x \to 1} \dfrac{\ln(x^4+1)-1}{x-2}\text{.}\)
Section 1.6 Limits
Subsection 1.6.1 Recommended Tutorials
Before starting on these exercises, you should familiarize yourself with the material covered in the following tutorial:
Subsection 1.6.2 Introduction
In this activity, you will use the
limit() command to evaluate the limit of an expression or function. This will also include one-sided limits and limits at infinity.
Exercises 1.6.3 Exercises
2.
Limits are quite useful for determining the behavour of a function around a discontinuity.
Aside
(a)
Assign the function
\begin{equation*}
f(x)=\dfrac{|x|}{x}
\end{equation*}
using the assignment operator
:=.(b)
(c)
Using three different
limit() commands, evaluate left-hand limit, the right-hand limit, and the two-sided limit as \(x\) approaches zero.
3.
Suppose we wish to determine the behavour of
\begin{equation*}
g(x)=\dfrac{x^2+x}{\sqrt{x^3+x^2}}
\end{equation*}
on either side of the discontinuity at \(x=0\text{.}\)
(a)
Start by assigning the function. Then, using the
plot() command on the interval x=-2..2, notice whether the left- and right-hand limits appear to be equal.
(b)
Use two
limit() commands to calculate the limits
\begin{equation*}
\lim_{x \to 0^-}g(x) \text{ and } \lim_{x \to 0^+}g(x)\text{.}
\end{equation*}
4.
Unless otherwise specified, the
limit() command will always default to a two-sided limit. Remember that the two-sided limit exists if and only if both one-sided limits exist and are equal.
(a)
Plot the graph of
\begin{equation*}
\frac{\sin(t)}{\sin(\pi t)}
\end{equation*}
and observe the behavour of the function around \(t=0\text{.}\) Does it appear that
\begin{equation*}
\lim_{t \to 0^-}\frac{\sin(t)}{\sin(\pi t)} \text{ and } \lim_{t \to 0^+}\frac{\sin(t)}{\sin(\pi t)}
\end{equation*}
both exist? Do they appear to be equal?
Hint 2.
(b)
Evaluate \(\displaystyle\lim_{t \to 0}\frac{\sin(t)}{\sin(\pi t)}\) using the
limit() command and confirm whether the two-sided limit exists.
5.
Calculate \(\displaystyle\lim_{x \to \infty} \sqrt{x+1}-x\text{.}\)
6.
If you wish to explore all of the limit laws and how they apply to a challenging limit problem, you can see each step applied individually using the Limit Methods tutor. In this exercise, you will get to see a step-by-step method for evaluating
\begin{equation*}
\lim_{x \to -\infty} \sqrt{x^2+x+1}+x\text{.}
\end{equation*}
(a)
Open the Limit Methods tutor and type in the function
sqrt(x^2+x+1)+x. Set the variable to x at -infinity and click Start to initialize the tutor. Click "Next Step" twice to see how Maple rewrites the expression by obtaining a common denominator and dividing by the highest power.
Hint.
An example involving the Limit Methods tutor can be found in Section 11.5.
(b)
Finish evaluating the limit by using the buttons on the right to apply a specific limit law, or click "All Steps" to watch Maple apply all of the necessary laws.
