Introduction to Limits in Calculus: Concepts, Examples, and Applications

Introduction to Limits

Motivation: Average vs. Instantaneous Velocity

Limits are fundamental in calculus, providing a way to rigorously define instantaneous rates of change, such as velocity. The distinction between average and instantaneous velocity motivates the need for limits.

• Position function: $x(t)$ gives the position of a particle at time $t$.

• Instantaneous velocity: $v(t)$ is the velocity of the particle at time $t$.

• Average velocity over interval $[t_1, t_2]$: $\bar{v} = \frac{\Delta x}{\Delta t} = \frac{x(t_2) - x(t_1)}{t_2 - t_1}$

Example 1: Estimating Instantaneous Velocity

Suppose $x(t) = 16t^2$. Estimate the instantaneous velocity at $t = 2$.

• Use average velocity over intervals approaching $t = 2$:

• For $[2, 2.1]$: $\bar{v} = \frac{x(2.1) - x(2)}{2.1 - 2} = \frac{16(2.1)^2 - 16(2)^2}{0.1} = 66.56$

• For smaller intervals, the estimate improves:

Interval	$\Delta t$	$\bar{v}$
[2, 2.1]	0.1	66.56
[2, 2.01]	0.01	64.16
[2, 2.001]	0.001	64.016
[1.9, 2]	0.1	61.44

As $\Delta t$ gets smaller (closer to 0), $\bar{v}$ approaches 64.

Limit Notation and Instantaneous Velocity

The process above leads to the concept of a limit:

• Limit notation: $\lim_{\Delta t \to 0} \bar{v} = 64$

• This expresses that as the interval shrinks, the average velocity approaches the instantaneous velocity.

Exact Calculation Using Limits

We can calculate the instantaneous velocity exactly using limits and symbolic algebra.

• Let $h > 0$, consider the average velocity over $[2, 2+h]$:

• $\bar{v} = \frac{x(2+h) - x(2)}{h}$

• Given $x(t) = 16t^2$:

• $x(2+h) = 16(2+h)^2 = 16(4 + 4h + h^2) = 64 + 64h + 16h^2$

• $x(2) = 64$

• $\bar{v} = \frac{(64 + 64h + 16h^2) - 64}{h} = \frac{64h + 16h^2}{h} = 64 + 16h$

• As $h \to 0$, $\bar{v} \to 64$

General Definition of Limits

The formal definition of a limit is:

• $\lim_{x \to a} f(x) = L$ means the values of $f(x)$ can be made arbitrarily close to $L$ as $x$ approaches $a$.

Estimating Limits Numerically and Graphically

Example 2: Numerical Estimation

Estimate $\lim_{x \to 2} \frac{x^2 - 2x - 3}{x - 2}$ using a table of values:

x	$f(x)$
2.5	3.99
2.1	3.91
2.01	3.901
1.99	4.099
1.9	4.09
1.5	4.6

This suggests the limit is 4.

Example 3: Graphical Estimation

Use a graph to estimate $\lim_{x \to 2} \frac{x^2 - 2x - 3}{x - 2}$.

• Graph of $f(x) = \frac{x^2 - 2x - 3}{x - 2}$ is undefined at $x = 2$.

• For $x$ near 2, $f(x)$ approaches 4.

• Graph visually confirms the limit is 4.

Example 4: Algebraic Calculation of Limits

Calculate $\lim_{x \to 2} \frac{x^2 - 2x - 3}{x - 2}$ using algebra:

• Factor numerator: $x^2 - 2x - 3 = (x - 3)(x + 1)$

• $\frac{(x - 3)(x + 1)}{x - 2}$

• Plug $x = 2$ into the simplified expression (after canceling if possible):

• $\lim_{x \to 2} \frac{(2 - 3)(2 + 1)}{2 - 2}$ is undefined, but for $x \neq 2$, the function approaches 4.

• Therefore, $\lim_{x \to 2} \frac{x^2 - 2x - 3}{x - 2} = 4$

One-Sided Limits

Definitions and Comparison to General Limits

• Left-sided limit: $\lim_{x \to a^-} f(x) = L$ is the limit as $x$ approaches $a$ from the left.

• Right-sided limit: $\lim_{x \to a^+} f(x) = L$ is the limit as $x$ approaches $a$ from the right.

• Both definitions mean the values of $f(x)$ can be made arbitrarily close to $L$ as $x$ approaches $a$ from the respective side.

Example 5: Finding Limits from a Graph

Given a graph of $f(x)$, find the following limits:

• (a) $\lim_{x \to 3^-} f(x) = 3$

• (b) $\lim_{x \to 3^+} f(x) = 1$

• (c) $\lim_{x \to 3} f(x)$ DNE (Does Not Exist)

• (d) $\lim_{x \to 1} f(x) = 2.5$

• (e) $\lim_{x \to -1} f(x) = -1$

Key Point: The two-sided limit exists only if both one-sided limits are equal.

Additional info: These notes provide foundational concepts for understanding limits, including their application to velocity, numerical and graphical estimation, algebraic calculation, and the distinction between one-sided and two-sided limits. Mastery of these topics is essential for further study in calculus, particularly in differentiation and continuity.