Introduction to Limits: Concepts, Definitions, and Applications

Unit 1: Introduction to Limits

Overview of Limits

Limits are a foundational concept in calculus, essential for understanding instantaneous velocity, the slope of a tangent line, and the behavior of functions near specific points. This section introduces the formal and intuitive definitions of limits, their graphical interpretation, and cases where limits do not exist.

• Instantaneous velocity and slope of a tangent line are key applications of limits.

• Limits help analyze jumps, oscillations, and unbounded behavior in functions.

Limits and Instantaneous Velocity

Average vs. Instantaneous Velocity

The average velocity of an object over a time interval is the change in position divided by the change in time. The instantaneous velocity is the velocity at a single moment, found by taking the limit as the time interval approaches zero.

• Average velocity formula: $\frac{f(a+h) - f(a)}{h}$

• Instantaneous velocity: $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

• Example: For an object thrown upward with $v_0 = 96$ ft/s, $s(t) = 96t - 16t^2$ gives position at time $t$.

Slope of the Tangent Line

The slope of the tangent line to a curve at a point is the limit of the slopes of secant lines as the interval shrinks to zero. This is a geometric interpretation of the derivative.

• The tangent slope at $x = a$ is $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$.

Definitions of Limits

Preliminary Definition

Suppose $f(x)$ is defined for all $x$ near $a$ except possibly at $a$. The limit $\lim_{x \to a} f(x) = L$ means $f(x)$ gets arbitrarily close to $L$ as $x$ gets arbitrarily close to $a$ (from either side).

• We write $\lim_{x \to a} f(x) = L$ if $f(x)$ approaches $L$ as $x$ approaches $a$.

• The value of $f(a)$ does not affect the limit.

Finding Limits Graphically and Numerically

• Limits can be estimated from graphs by observing the $y$-value as $x$ approaches $a$ from both sides.

• Numerical tables can be used to approximate limits by evaluating $f(x)$ for values of $x$ close to $a$.

One-Sided Limits

Left-Sided and Right-Sided Limits

One-sided limits consider the behavior of $f(x)$ as $x$ approaches $a$ from only one direction.

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

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

• A two-sided limit exists only if both one-sided limits exist and are equal.

Theorem: Relationship Between One-Sided and Two-Sided Limits

The two-sided limit exists if and only if both one-sided limits exist and are equal:

• $\lim_{x \to a} f(x) = L$ if and only if $\lim_{x \to a^-} f(x) = L$ and $\lim_{x \to a^+} f(x) = L$

Limits Involving Absolute Value

Piecewise Definition of Absolute Value

The absolute value function can be defined piecewise, which is useful for evaluating limits involving $|x|$.

• $|x| = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases}$

Different Right and Left Behavior

Some functions have different limits from the left and right at a point, indicating a jump or discontinuity.

Cases Where Limits Fail to Exist

Types of Non-Existence

• Jump discontinuity: Left and right limits are not equal.

• Unbounded behavior: Function approaches infinity or negative infinity.

• Oscillating behavior: Function does not settle to a single value as $x$ approaches $a$.

Summary Table: Types of Limit Behavior

Case	Description	Limit Exists?
Jump	Left and right limits not equal	No
Unbounded	Function approaches infinity	No
Oscillating	Function fluctuates without settling	No

Key Takeaways

• Limits describe the behavior of functions as inputs approach a specific value.

• They are essential for defining derivatives and understanding continuity.

• Limits can be estimated graphically, numerically, or analytically.

• One-sided limits help analyze discontinuities and piecewise functions.

• Limits may fail to exist due to jumps, unboundedness, or oscillation.