5.1 Introduction to Limits and Derivatives
1. The Limit $\lim_{x \to a} f(x)$
Consider the function $f(x) = 2x + 3$.
Let us investigate how the function behaves at $x=2$. Clearly $f(2) = 7$. But what happens when $x$ is very close to 2?
| $x$ approaches $2^-$ (from values less than 2) |
|
|---|---|
| $x$ | $f(x)$ |
| 1.9 | 6.8 |
| 1.99 | 6.98 |
| 1.999 | 6.998 |
| $x$ approaches $2^+$ (from values greater than 2) |
|
|---|---|
| $x$ | $f(x)$ |
| 2.1 | 7.2 |
| 2.01 | 7.02 |
| 2.001 | 7.002 |
- If $x \to 2^-$ then $f(x) \to 7$
- If $x \to 2^+$ then $f(x) \to 7$
Thus in general, if $x$ tends to 2, $f(x)$ tends to 7. In order to express this fact we write:
and say that: the limit of $f(x)$, as $x$ tends to 2, is 7.
Remark on Side Limits
In fact, for the left approach we write $\lim_{x \to 2^-} f(x) = 7$, while for the right approach we write $\lim_{x \to 2^+} f(x) = 7$. These are mathematically called side limits. If the side limits are strictly equal, then:
In this continuous example, $\lim_{x \to 2} f(x) = 7$ which in fact equates directly to $f(2)$. Similarly, $\lim_{x \to 3} f(x) = 9$ which is $f(3)$. In general, for standard polynomial curves: $\lim_{x \to a} f(x) = f(a)$.
The situation occurs very often; however, this is not universally the case (otherwise the limit would be nothing more than a simple algebraic substitution!).
EXAMPLE 1
Let's see a definitive case where the limit is not a simple substitution! Consider the trigonometric function $f(x) = \dfrac{\sin x}{x}$. It is structurally not defined at $x=0$. However, we will find (informally) the limit $\lim_{x \to 0} \dfrac{\sin x}{x}$.
Although $f(0)$ fundamentally does not exist, it seems by graphical observation that the limit exactly at $x=0$ is 1. Let's approach $x=0$ by actively using our GDC:
| $x$ approaches $0^-$ (from values less than 0) |
|
|---|---|
| $x$ | $f(x)$ |
| -0.1 | 0.998334 |
| -0.01 | 0.999983 |
| -0.001 | 0.999999 |
| $x$ approaches $0^+$ (from values greater than 0) |
|
|---|---|
| $x$ | $f(x)$ |
| 0.1 | 0.998334 |
| 0.01 | 0.999983 |
| 0.001 | 0.999999 |
- If $x \to 0^-$ then $f(x) \to 1$
- If $x \to 0^+$ then $f(x) \to 1$
The functional limit when $x$ tends to 0 strictly converges to 1. We write:
2. Vertical and Horizontal Asymptotes
The limit can explicitly evaluate to $+\infty$ or $-\infty$. Let $f(x) = \dfrac{1}{x}$.
At $x=0$, we only have extreme diverging side-limits:
In fact, these physical results justify that $x=0$ operates as a vertical asymptote. In general: If $\lim_{x \to a^+} f(x) = \pm\infty$ or $\lim_{x \to a^-} f(x) = \pm\infty$ (or both), we state formally that $x=a$ is a vertical asymptote.
We also define boundaries of the form $\lim_{x \to +\infty} f(x)$ or $\lim_{x \to -\infty} f(x)$. We observe the behavior of the functional curve when $x$ physically approaches $+\infty$ (large positive) or $-\infty$ (large negative).
In our example $f(x) = \dfrac{1}{x}$ (see graph above):
For example, if $x=1,000,000$ or $-1,000,000$, then $y$ strictly decays close to 0. These analytical results identically justify that $y=0$ functions as a horizontal asymptote. In general: If $\lim_{x \to +\infty} f(x) = a$ or $\lim_{x \to -\infty} f(x) = a$ (or both), we state formally that $y=a$ is a horizontal asymptote.
EXAMPLE 2
Consider the rational function $f(x) = \dfrac{x-3}{x-2}$.
- $x=2$ is a vertical asymptote. The mathematical formal justification is that $\lim_{x \to 2^-} f(x) = +\infty$ and $\lim_{x \to 2^+} f(x) = -\infty$.
- $y=1$ is a horizontal asymptote. The mathematical formal justification is that $\lim_{x \to +\infty} f(x) = 1$ and $\lim_{x \to -\infty} f(x) = 1$.
EXAMPLE 3
$y=10$ is a horizontal asymptote for $f(x)$. The explicit formal justification evaluates to $\lim_{x \to +\infty} f(x) = 10$.
$x=4$ is a vertical asymptote for $g(x)$. The explicit formal justification evaluates to $\lim_{x \to 4^+} g(x) = -\infty$.
EXAMPLE 4
Look at an interesting limit parameter that constructs the fundamental irrational number $e = 2.7182818...$
Investigate (informally) the limit boundary $\lim_{x \to +\infty} \left(1 + \dfrac{1}{x}\right)^x$.
| $x$ approaches $+\infty$ | |
|---|---|
| $x$ | $f(x)$ |
| 1000 | 2.7169239... |
| 1,000,000 | 2.7182804... |
| $10^{10}$ | 2.7182818... |
The resulting converging limit is in fact the physical number $e$. That strictly is,
3. Rate of Change (Gradient) in a Straight Line
Consider the linear curve $f(x) = 2x + 3$. Let us systematically pick the points $(1,5)$ and $(2,7)$ operating on the line.
Notice: when $x$ changes strictly from 1 to 2, then $y$ functionally changes from 5 to 7. Hence, the mathematically corresponding rate of change evaluates to:
We understand fundamentally that for a straight line, the rate of change between any two given points is consistently the same absolute value. For example, if we pick the distinct points $(0,3)$ and $(2,7)$:
This constant numerical value is specifically defined as the gradient of the line.
Next, we will verify that the gradient concept is not only isolated to straight lines but expands dynamically for other polynomial curves.
4. Rate of Change (Gradient) in a Curve
In a polynomial curve which physically diverges from a straight line, the isolated rate of change between any two discrete points is explicitly not always identical. For example, in $f(x) = x^2$:
- Rate of change from $x=1$ to $x=2$: $\quad \dfrac{\Delta y}{\Delta x} = \dfrac{f(2)-f(1)}{2-1} = \dfrac{4-1}{1} = \mathbf{3}$
- Rate of change from $x=1$ to $x=3$: $\quad \dfrac{\Delta y}{\Delta x} = \dfrac{f(3)-f(1)}{3-1} = \dfrac{9-1}{2} = \mathbf{4}$
However, we can systematically measure the "instantaneous" rate of change exactly localized at any point $P(x,y)$ on the curve boundary. This specific vector will definitively evaluate the gradient at point $P$.
The random coordinate point $P$ has the parametric form $P(x, x^2)$. Let $Q$ establish a point incredibly close to $P$, say positioned at $x+h$: $Q(x+h, (x+h)^2)$.
Rate of mathematical change scaling from $P$ to $Q$ evaluates structurally as:
By actively letting dynamic interval $h$ tend to effectively 0, we isolate and finalize the exact gradient at isolated point $P$:
Hence, the absolute gradient mapped at any independent point $x$ on the exponential curve evaluates natively to $m=2x$.
The Derivative Function
In the subsequent geometric diagram we effectively provide the gradient mapped discretely at some independent points of the specific curve $y=x^2$:
- gradient exactly at $x=-1$: $\quad m = -2$
- gradient exactly at $x=0$: $\quad m = 0$
- gradient exactly at $x=1$: $\quad m = 2$
- gradient exactly at $x=2$: $\quad m = 4$
- gradient exactly at $x=3$: $\quad m = 6$
- gradient exactly at $x=-1$: $\quad m = -2$
- gradient exactly at $x=0$: $\quad m = 0$
- gradient exactly at $x=1$: $\quad m = 2$
- gradient exactly at $x=2$: $\quad m = 4$
- gradient exactly at $x=3$: $\quad m = 6$
Thus, for the base functional curve $f(x) = x^2$, a definitively new function explicitly denoted $f'(x)$ is established structurally; it mathematically maps the gradient vector at any isolated domain variable $x$:
Thus, for a purely calculated example, the local gradient existing at $x=3$ evaluates structurally to: $f'(3) = 6$.
The newly defined boundary function $f'(x)$, strictly corresponding mathematically to initial condition $f(x)$, is explicitly classified as the derivative of $f(x)$.