Definitions and Evaluation of Limits in Calculus

Definitions of Limits

Limit of a Function (Preliminary)

The concept of a limit is fundamental in calculus and describes the behavior of a function as its input approaches a particular value. Formally, if a function f is defined for all x near a value a (except possibly at a), and if f(x) becomes arbitrarily close to L as x approaches a, we write:

$\lim_{x \to a} f(x) = L$

• Key Point: The limit describes the value that f(x) approaches, not necessarily the value of f(a).

• Application: Limits are used to define continuity, derivatives, and integrals in calculus.

Example: If f(x) gets closer to L as x approaches a from both sides, then $\lim_{x \to a} f(x) = L$.

Evaluating Limits Graphically and Numerically

Using Graphs

Limits can be estimated by observing the behavior of a function's graph near the point of interest.

• Key Point: The value of the limit depends on the values of f(x) near a, not necessarily at a.

• Example: Use the graph to estimate $\lim_{x \to 1} f(x)$ by observing the y-values as x approaches 1 from both sides.

Using Tables

Numerical tables can be used to approximate limits by evaluating f(x) at values increasingly close to a.

• Key Point: As x gets closer to a, the values of f(x) should approach the limit.

x	f(x)
0.9	0.531316
0.99	0.503312
0.999	0.500332
1.001	0.499668
1.01	0.496688
1.1	0.481125

Example: For $f(x) = \frac{\sqrt{x} - 1}{x - 1}$, as x approaches 1, the values of f(x) approach 0.5.

One-Sided Limits

Definitions

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

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

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

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 $\lim_{x \to a^+} f(x) = L$

Finding Limits from a Graph

Graphical Evaluation

To find limits from a graph, observe the y-values as x approaches the point of interest from the left and right.

• Key Point: If the left and right limits are not equal, the two-sided limit does not exist.

• Example: For a piecewise function, check the graph at the point of interest to determine if the function approaches the same value from both sides.

Practice Problems

Evaluating Limits Graphically

• Sketch the graph of the function and use it to estimate $f(a)$, $\lim_{x \to a^-} f(x)$, $\lim_{x \to a^+} f(x)$, and $\lim_{x \to a} f(x)$.

• For piecewise functions, evaluate limits at points where the definition changes.

Example: For $f(x) = \begin{cases} x^2 + 1 & \text{if } x \leq -1 \\ 3 & \text{if } x > -1 \end{cases}$, estimate the limits as $x$ approaches $-1$ from both sides.

Numerical and Algebraic Evaluation

• Use tables to estimate limits for functions with removable discontinuities.

• Apply algebraic simplification to find limits, such as factoring and canceling terms.

Example: For $f(x) = \frac{x^2 + x - 2}{x - 1}$, factor the numerator and simplify before evaluating the limit as $x \to 1$.

Explaining Limit Statements

True/False and Counterexamples

• Not all limits are found by direct substitution; some require simplification or graphical analysis.

• If $f(a)$ is undefined, the limit $\lim_{x \to a} f(x)$ may still exist.

Example: $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$ exists even though $f(3)$ is undefined, because the expression simplifies to $x + 3$ for $x \neq 3$.

Special Limits and Calculator Use

Limits Involving Trigonometric and Radical Functions

• Use calculators or tables to estimate limits for complex functions.

• For $\lim_{h \to 0} \frac{\sin h}{h}$, the value is 1 (important in derivative definitions).

Example: For $\lim_{x \to \frac{\pi}{2}} \cot x = 0$, use the graph of $y = \cot x$ to confirm.

Summary Table: Types of Limits

Type	Notation	Description
Two-sided	$\lim_{x \to a} f(x)$	Approaches from both sides
Right-sided	$\lim_{x \to a^+} f(x)$	Approaches from the right
Left-sided	$\lim_{x \to a^-} f(x)$	Approaches from the left

Additional info: These notes cover the foundational concepts of limits, including graphical, numerical, and algebraic approaches, as well as one-sided and two-sided limits. Practice problems and examples are provided to reinforce understanding and application.