Limits and Limit Laws in Calculus: Study Notes and Examples

Limits and Continuity

Introduction to Limits

Limits are a foundational concept in calculus, describing the behavior of functions as inputs approach a specific value. Understanding limits is essential for studying derivatives, integrals, and continuity.

• Definition: The limit of a function f(x) as x approaches c is the value that f(x) gets closer to as x gets arbitrarily close to c.

• Notation: $\lim_{x \to c} f(x)$

• Key Point: Limits may exist even if the function is not defined at that point.

Limit Laws (Theorem 1)

Limit laws provide rules for evaluating limits of combinations of functions. If $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} g(x) = M$, then:

• Sum Rule: $\lim_{x \to c} [f(x) + g(x)] = L + M$

• Difference Rule: $\lim_{x \to c} [f(x) - g(x)] = L - M$

• Constant Multiple Rule: $\lim_{x \to c} [k f(x)] = kL$ (where k is a constant)

• Product Rule: $\lim_{x \to c} [f(x) g(x)] = LM$

• Quotient Rule: $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L}{M}$ (if $M \neq 0$)

• Power Rule: $\lim_{x \to c} [f(x)]^n = L^n$ (n is a positive integer)

• Root Rule: $\lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{L}$ (if n is even, require $L \geq 0$)

Examples Using Limit Laws

• Example 1: $\lim_{x \to 2} [2f(x) + 3g(x)]$ where $\lim_{x \to 2} f(x) = 5$, $\lim_{x \to 2} g(x) = -7$ $= 2 \times 5 + 3 \times (-7) = 10 - 21 = -11$

• Example 2: $\lim_{x \to 2} \frac{-4p(x) + 5r(x)}{s(x)}$ where $\lim_{x \to 2} p(x) = 4$, $\lim_{x \to 2} r(x) = 0$, $\lim_{x \to 2} s(x) = -3$ $= \frac{-4 \times 4 + 5 \times 0}{-3} = \frac{-16}{-3} = \frac{16}{3}$

Indeterminate Forms and Special Limits

Some limits result in indeterminate forms, such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$. These require further analysis, often using algebraic manipulation or special theorems.

• Example: $\lim_{x \to 0} \frac{x}{|x|}$ For $x > 0$, $\frac{x}{|x|} = 1$; for $x < 0$, $\frac{x}{|x|} = -1$. The left and right limits are not equal, so the limit does not exist.

Greatest Integer Function and Limits

The greatest integer function $\lfloor x \rfloor$ returns the largest integer less than or equal to x. Limits involving this function can be discontinuous or undefined at integer points.

• Example: $\lim_{x \to 0} \lfloor x \rfloor = 0$

• Example: $\lim_{x \to 0} \frac{\lfloor x \rfloor}{x}$ As $x \to 0^+$, $\lfloor x \rfloor = 0$, so the limit is 0. As $x \to 0^-$, $\lfloor x \rfloor = -1$, so the limit diverges to $-\infty$. The two-sided limit does not exist.

Special Trigonometric Limits Near Zero

Key Trigonometric Limits

Trigonometric functions have important limits near zero that are frequently used in calculus.

• $\lim_{x \to 0} \frac{\sin x}{x} = 1$

• $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$

Examples and Applications

• Example: $\lim_{x \to 0} \frac{1 - \cos x}{x}$ Using the special limit above, the answer is 0.

• Example: $\lim_{x \to 0} \frac{1 - \cos x}{\sin x}$ As $x \to 0$, $\frac{1 - \cos x}{\sin x} \to 0$

• Example: $\lim_{x \to 0} \frac{1 - \cos x}{x^2}$ Using the Taylor expansion, $1 - \cos x \approx \frac{x^2}{2}$ for small $x$, so the limit is $\frac{1}{2}$.

The Sandwich (Squeeze) Theorem

Theorem Statement

The Sandwich Theorem (also called the Squeeze Theorem) is used to find limits of functions that are "squeezed" between two other functions with known limits.

• If $g(x) \leq f(x) \leq h(x)$ for all $x$ near $c$ (except possibly at $c$), and $\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$, then $\lim_{x \to c} f(x) = L$.

Example Using the Sandwich Theorem

• Example: $\lim_{x \to 0} x^2 \sin \left( \frac{1}{x} \right)$ Since $-1 \leq \sin \left( \frac{1}{x} \right) \leq 1$, $-x^2 \leq x^2 \sin \left( \frac{1}{x} \right) \leq x^2$ Both $\lim_{x \to 0} -x^2 = 0$ and $\lim_{x \to 0} x^2 = 0$, so by the Sandwich Theorem: $\lim_{x \to 0} x^2 \sin \left( \frac{1}{x} \right) = 0$

Graphical Interpretation of Limits

Oscillating Functions and Limits

Some functions oscillate infinitely as x approaches a point, making the limit undefined. Graphs can help visualize such behavior.

• Example: $y = \sin \left( \frac{1}{x} \right)$ as $x \to 0$ oscillates between -1 and 1 infinitely often, so the limit does not exist.

Summary Table: Limit Laws

Rule	Formula
Sum Rule	$\lim_{x \to c} [f(x) + g(x)] = L + M$
Difference Rule	$\lim_{x \to c} [f(x) - g(x)] = L - M$
Constant Multiple Rule	$\lim_{x \to c} [k f(x)] = kL$
Product Rule	$\lim_{x \to c} [f(x) g(x)] = LM$
Quotient Rule	$\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L}{M}$ (if $M \neq 0$)
Power Rule	$\lim_{x \to c} [f(x)]^n = L^n$
Root Rule	$\lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{L}$

Additional info:

• Some examples and explanations were expanded for clarity and completeness.

• Graphical examples were interpreted based on standard calculus knowledge.