Motivation

  1. Building Models: We explored how calculus serves as a powerful tool to create mathematical models. Newton’s gravitational model demonstrated that calculus is the bedrock upon which many university courses build their mathematical frameworks.

  2. Precise Communication: Calculus provides us with a universal language to communicate abstract concepts effectively. Without this language, the exchange of ideas can be vague, hindering collaboration and understanding.

  3. Cultural Significance: From everyday technologies like smartphones to cutting-edge advancements in artificial intelligence, calculus is the unifying foundation behind many innovations. Regardless of one’s specific field of study, learning calculus contributes to a shared cultural background among scientists and scholars.

Instantaneous velocity and the slope of a tangent line to a curve

Let $x = p(t)$ represent a position function, describing the location of an object under observation at a specific time, $t$. To determine the average velocity between two time points, $t_1$ and $t_2$, we use the following formula:

\[v_{t_1t_2} = \frac{p(t_1) - p(t_2)}{t_1 - t_2}.\]

When seeking the instantaneous velocity at time $t_1$, we consider the scenario where time $t_2$ approaches $t_1” as close as we wish.”

Alternatively, when we visualize the function $x = p(t)$ graphically on a plane, $v_{t_1t_2}$ represents the slope of the secant line connecting the points $(t_1, p(t_1))$ and $(t_2, p(t_2))$. As we allow $t_2$ to approach ever closer to $t_1$, these secant lines progressively approximate the tangent lines to the curve $x = p(t)$ at the point $(t_1, p(t_1))$.

In this light, we can conclude that the concept of instantaneous velocity in physics aligns with the mathematical notion of the slope of the tangent line to a curve at a specific point.

Try to make it precise

The explanation above highlights a key motivation for rigorously defining the concept of limit. Our initial definition provides an intuitive understanding of limits.

Definition. Suppose $f(x)$ is defined when $x$ is near the number $a$ (This means that $f$ is defined around $a$, except possibly at $a$ itself.). Then we write \(\lim_{x\to a}f(x)=L\) and say “the limit of $f(x)$, as $x$ approaches $a$, equals $L$” if we can make the values of $f(x)$ arbitrarily close to $L$ by restricting $x$ to be sufficiently close to $a$ but not equal to $a$.

When we consider both the left and right limits, we define a limit $\displaystyle\lim_{x\to a} f(x)$ to exist if and only if both the left and right limits exist and satisfy the condition: \(\lim_{x\to a^{-}}f(x) = \lim_{x\to a^+}f(x)\)

Understanding limits through numbers

Let us see the following examples.

Example 1. $\displaystyle\lim_{t\to 0}\frac{\sqrt{t^2+9}-3}{t^2}=1.66….$

f(t) = ((t^2+9)^(0.5)-3)/ t^2 label = [('t','f(t)')] result = [] for x in [10^(-i) for i in range(1,10)]: result.append((x.n(100), f(x.n(100)))) #Note the difference between f(x.n(100)) and f(x).n(100) printout = label + result table(printout, header_row=True, frame=True)

From this example, we can numerically determine the limit is $1.\bar{6}$.

Example 2. We have explored the limit of $\sin(1/t)$ as $t$ approaches $0” and have observed that there is no discernible pattern or limit.

f(t) = sin(1/t) label = [('t','f(t)')] result = [(x.n(),f(x.n())) for x in [10^(-i) for i in range(1,10)]] printout = label + result table(printout, header_row=True, frame=True)

Similarly, when we examine the limit of $t^{0.001}\sin(1/t)$, we find that numerical results do not provide a conclusive answer.

f(t) = t^(0.001)*sin(1/t) label = [('t','f(t)')] result = [(x.n(),f(x.n())) for x in [10^(-i) for i in range(1,10)]] printout = label + result table(printout, header_row=True, frame=True)

As it turn out, the first limit does not exist(DNE), while the second limit approaches zero. This highlights that relying solely on numerical methods may not always yield a definitive or accurate result.

DNE

When we declare that a limit does not exist, it may signify one of two possibilities: either we possess inadequate information to ascertain a limit, or we can articulate a precise description of the limit’s behavior.

For instance, consider the limit of $1/x$ as $x$ approaches $0$ from the right. In this case, the limit doesn’t converge to a specific real number, so we denote it as “DNE” (does not exist). However, we can still provide a specific characterization of its behavior, noting that it approaches positive infinity \(\lim_{x\to 0^{+}}\frac{1}{x} = \infty.\) Similarly, we have \(\lim_{x\to 0^-}\frac{1}{x} = -\infty.\)

plot(1/x)

Take a look at the graph. Notice that the line $x=0$ (the $y$-axis) serves as a vertical asymptote for the curve $y=1/x$. This relationship provides a geometric interpretation for this type of limit.

Definition. We say $x=a$ is a vertical asysmptote of a curve $y=f(x)$ if one of the following is true:

\[\lim_{x\to a^+}f(x)=\pm\infty\text{ or }\lim_{x\to a^-}f(x)=\pm\infty.\]

Remark. “Diverging to infinity” is a category within the classification of “does not exist” (DNE)