Lab 3: Limits II



This lab considers the topic of limits again, but in a more mathematically rigorous manner. In the previous lab, you thought of the statement \( \lim\limits_{x\to c}f(x) = L\) as a prediction of a \(y\)-value based on the trend of the values \(f(x)\) for inputs \(x\) near \(c\). The goal here is to make the idea of “trend of values” more precise. We want those values \(f(x)\) to “stay close” to \(L\) when we “zoom near enough” to the point \( x=c \). To get a deeper understanding of what this means, we need to quantify the notions “near enough” and “stay close.”

Formal Approach to Limits

Recall the formal definition of limit from §2.5 of the textbook:

Definition. Let \( f(x)\) be defined for all \( x \ne c\) over an open interval \( (a,b)\) containing \( c \). Let \( L \) be a real number. Then
\[ \lim_{x \to c} f(x) = L \] means that, for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that, whenever \( c-\delta < x < c \) or \(c < x < c+ \delta \), one has \( L-\varepsilon < f(x) < L+\varepsilon \).

Intuitively, \( \varepsilon \) controls how far we “zoom in” in the \( y \)-direction while \( \delta \) controls how far we “zoom in” in the \( x \)-direction. So the definition above says that, no matter how far we zoom in in the \( y \)-direction, we can zoom in in the \( x \)-direction far enough so that the function is contained entirely in the box (except possibly at \( x = c\)). Graphically, this is represented by the function “exiting” the box only through the sides (and not the top or bottom).

Formally, when checking this definition, we want to answer the following question: Is it possible to select \( \delta \) in terms of \( \varepsilon \) so that the values \( f(x) \) will always be between \( L-\varepsilon \) and \(L+\varepsilon\) when the input \(x\) falls within the zooming window—that is, when \( c-\delta < x < c+\delta \) (except possibly at \( x = c \))? Is this possible for all \( \varepsilon > 0 \)?

The exercises in this lab investigate the above question. They deploy a SageMath demo, which differs from the demo from Lab 2: It replaces the zoom level with a choice of \( \varepsilon \) and a choice of \( \delta \). The demo also allows us to make a choice for the value \(L\) of the limit, and then sets the zooming window to the rectangle defined by the inequalities
\[ c-\delta \le x \le c + \delta, \qquad L-\varepsilon \le y \le L + \varepsilon. \] Note that the zooming window (right display) shows exactly what is in the red box in the left display, although the aspect ratio may not be equal.


1. Using the functions and \( L \) values given by your instructor, for each given \( \varepsilon \), find an appropriate \( \delta \) so that, whenever \( 1-\delta < x < 1 \) or \(1 < x < 1+ \delta \), \( L-\varepsilon < f(x) < L+\varepsilon \). Try to make \( \delta \) as large as possible.

2. Using the function F5 and \( L = -0.5 \), complete the following.

(a) Set \( \varepsilon = 0.05 \) and \( \delta = 0.05 \). Produce a printout of the plot and attach it to your lab report. From the plot, predict the value of \( \lim\limits_{x\to 3}f(x) \).

(b) For each given \( \varepsilon \), try to find an appropriate \( \delta \) so that, whenever \( 3-\delta < x < 3 \) or \(3 < x < 3 + \delta \), \( L-\varepsilon < f(x) < L+\varepsilon \). Try to make \( \delta \) as large as possible. If no such \( \delta \) exists, write “??”.

(c) Your observation in (a) suggests that \( \lim\limits_{x\to 3}f(x) \neq L \). How does your answer to part (b) confirm this?

3. Using the function and the \( L \) value given by your instructor, complete the following. Note that the value of \( c \) below is determined by the choice of function.

(a) Set \( \varepsilon = 0.05 \) and \( \delta = 0.05 \). Produce a printout of the plot and attach it to your lab report. What kind of discontinuity does \( f(x) \) appear to have at \( x = c \)? What does that tell you about \( \lim\limits_{x \to c} f(x) \)?

(b) For each given \( \varepsilon \), try to find an appropriate \( \delta \) so that, whenever \( c-\delta < x < c \) or \(c < x < c + \delta \), \( L-\varepsilon < f(x) < L+\varepsilon \). Try to make \( \delta \) as large as possible. If no such \( \delta \) exists, write “??”.

(c) Your observation in (a) suggests that \( \lim\limits_{x\to c}f(x) \neq L \). How does your answer to part (b) confirm this?

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