Calculus I: Limits, Continuity, and Tangent Lines – Study Notes

Limits and Their Evaluation

Definition and Basic Properties

Limits are a foundational concept in calculus, describing the behavior of a function as its input approaches a particular value. Understanding limits is essential for analyzing continuity, derivatives, and the behavior of functions near points of interest.

• Limit of a Function: The value that a function approaches as the input approaches a specific point.

• Notation: $\lim_{x \to a} f(x)$ denotes the limit of $f(x)$ as $x$ approaches $a$.

• One-Sided Limits: $\lim_{x \to a^-} f(x)$ (from the left), $\lim_{x \to a^+} f(x)$ (from the right).

Example:

• Evaluate $\lim_{x \to 2} (2x^2 - 3x + 6)$ by direct substitution: $2(2)^2 - 3(2) + 6 = 8 - 6 + 6 = 8$.

Special Limits and Techniques

• Trigonometric Limits: $\lim_{x \to 0} \frac{\sin x}{x} = 1$ is a key result.

• Factoring and Simplifying: For limits resulting in indeterminate forms ($0/0$), factor numerator and denominator to simplify.

• Squeeze Theorem: If $g(x) \leq f(x) \leq h(x)$ and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.

Example:

• Evaluate $\lim_{x \to 0} \sin\left(\frac{5}{x-5}\right)$ using substitution and trigonometric properties.

• Use the squeeze theorem to evaluate $\lim_{x \to 0^+} \sqrt{x} \sin\left(\frac{1}{x}\right)$, knowing $-\sqrt{x} \leq \sqrt{x} \sin\left(\frac{1}{x}\right) \leq \sqrt{x}$ and both bounds approach $0$as$x \to 0^+$.

Piecewise Functions and Continuity

Definition of Piecewise Functions

A piecewise function is defined by different expressions over different intervals of its domain. Analyzing limits and continuity at the boundaries of these intervals is crucial.

• Example Function:

  • $f(x) = \begin{cases} 3x-2, & x<2 \\ x^2, & 2 \leq x < 3 \\ x-4, & 3 \leq x \end{cases}$

Continuity and Discontinuity

• Continuous at a Point: $f$ is continuous at $x=a$ if $\lim_{x \to a} f(x) = f(a)$.

• Types of Discontinuity:

  • Jump Discontinuity: Left and right limits exist but are not equal.

  • Removable Discontinuity: Limit exists but does not equal the function value.

  • Infinite Discontinuity: Function approaches infinity near the point.

• Interval Notation: Used to describe where a function is continuous, e.g., $(a, b)$.

Example:

• Find $\lim_{x \to 2} f(x)$ for the piecewise function above by evaluating left and right limits.

• Describe the set of $x$-values where $f(x)$ is continuous using interval notation.

Finding Tangent Lines Using Limits

Definition and Process

The tangent line to a curve at a point represents the instantaneous rate of change (derivative) at that point. The slope of the tangent is found using the limit of the slope of secant lines.

• Secant Line: A line passing through two points on the curve.

• Tangent Line: A line that touches the curve at one point and has the same slope as the curve at that point.

• Formula for Slope:

  • $m = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

• Equation of Tangent Line:

  • $y = f(a) + m(x-a)$

Example:

• Find the slope of the tangent to $f(x)$ at $x=2$ using the limit definition.

• Write the equation of the tangent line at $P(2, f(2))$.

Summary Table: Types of Discontinuity

Type	Description	Example
Jump	Left and right limits exist but are not equal	Piecewise function with different values at boundary
Removable	Limit exists but does not equal function value	$f(x) = \frac{x^2-1}{x-1}$ at $x=1$
Infinite	Function approaches infinity near the point	$f(x) = \frac{1}{x}$ at $x=0$

Key Formulas and Theorems

• Limit Laws: Sum, product, quotient, and power rules for limits.

• Squeeze Theorem: Used to evaluate limits of functions bounded by two others.

• Continuity: $f$ is continuous at $x=a$ if $\lim_{x \to a} f(x) = f(a)$.

• Tangent Line Slope: $m = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

Additional info: These notes expand on the exam questions by providing definitions, examples, and context for each concept, ensuring a self-contained study guide for Calculus I students.