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

Limits and Continuity

Definition of a Limit

The limit of a function as x approaches a value describes the behavior of the function near that value, not necessarily at that value. Formally,

• If $\lim_{x \to a} f(x) = L$, then as x gets arbitrarily close to a (from either side), f(x) gets arbitrarily close to L.

Key Properties:

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

• Limits can be evaluated using direct substitution, factoring, rationalization, or L'Hospital's Rule (if indeterminate forms arise).

Example: $\lim_{x \to 2} (x^2 - 4)/(x - 2)$ can be evaluated by factoring numerator and canceling common terms.

Continuity

A function is continuous at a point x = a if:

• f(a) is defined

• $\lim_{x \to a} f(x)$ exists

• $\lim_{x \to a} f(x) = f(a)$

Types of Discontinuities:

• Removable: The limit exists, but the function is not defined or is defined differently at that point.

• Jump: The left and right limits exist but are not equal.

• Infinite: The function approaches infinity near the point.

Example: $f(x) = (x^2 - 1)/(x - 1)$ is discontinuous at x = 1, but the discontinuity is removable.

Average and Instantaneous Rate of Change

Average Speed

The average speed of an object over an interval [a, b] is given by:

• $\text{Average speed} = \frac{h(b) - h(a)}{b - a}$

where h(t) is the position function.

Instantaneous Speed (Derivative)

The instantaneous speed at time t = a is the derivative of the position function at that point:

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

This is the slope of the tangent line to the curve at t = a.

Tangent Line Equation

The equation of the tangent line to y = f(x) at x = a is:

• $y = f(a) + f'(a)(x - a)$

Example: For $h(t) = -16t^2 + 2880$, the instantaneous speed at t = 1 is $h'(1)$, and the tangent line at (1, h(1)) is $y = h(1) + h'(1)(t - 1)$.

Evaluating Limits: Techniques and Examples

Direct Substitution

• If $f(x)$ is continuous at $x = a$, then $\lim_{x \to a} f(x) = f(a)$.

Factoring and Canceling

• Factor numerator and denominator to cancel common terms before substituting.

Rationalization

• Multiply numerator and denominator by a conjugate to simplify expressions involving square roots.

Special Trigonometric Limits

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

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

L'Hospital's Rule

• If $\lim_{x \to a} \frac{f(x)}{g(x)}$ yields $0/0$ or $\infty/\infty$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$ (if the latter limit exists).

Examples of Limits

• $\lim_{x \to \pi/2} (\sin(x/2) - 3\cos(x))$

• $\lim_{x \to 0} \frac{\sin(5x)}{x}$

• $\lim_{x \to 4} \frac{\sqrt{x+5} - 3}{x - 4}$

• $\lim_{x \to 3} \frac{7x - 21}{x^2 - 9}$

• $\lim_{x \to -2} \frac{x + 1}{x^2 + 5x + 6}$

• $\lim_{x \to 1} \frac{7x - 3}{\sqrt{x^2 + 1} - 1}$

Each of these can be solved using the above techniques.

Interval Notation and Continuity

Interval Notation

• Used to describe sets of x-values where a function is continuous.

• Example: (a, b) means all x between a and b, not including endpoints.

Describing Continuity

• Identify intervals where the function is defined and has no discontinuities.

• Use limits to check for removable or non-removable discontinuities.

Intermediate Value Theorem (IVT)

The Intermediate Value Theorem states that if f(x) is continuous on [a, b] and k is any value between f(a) and f(b), then there exists c in (a, b) such that f(c) = k.

• Used to show the existence of roots in an interval.

Example: For $f(x) = x^3 - 5x^2 - 10x^2 + 15x + 30$, check if $f(x) = 0$ has a solution in [2, 3] by evaluating f(2) and f(3).

Absolute Value Inequalities and Limits

Absolute Value Inequality

• On [-1, 1], $-1 \leq \sin(x/2) \leq 1$

• Use this to evaluate limits involving absolute values and trigonometric functions.

Example: Use the squeeze theorem to evaluate $\lim_{x \to 0} (\cos(x/2))$.

Summary Table: Types of Discontinuities

Type	Description	Example
Removable	Limit exists, function not defined or defined differently at point	$f(x) = (x^2 - 1)/(x - 1)$ at x = 1
Jump	Left and right limits exist but are not equal	Piecewise function with different values on each side
Infinite	Function approaches infinity near the point	$f(x) = 1/(x - a)$ at x = a

Key Formulas

• Average rate of change: $\frac{f(b) - f(a)}{b - a}$

• Instantaneous rate of change (derivative): $f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$

• Tangent line: $y = f(a) + f'(a)(x - a)$

Additional info: The above notes are based on the exam questions, which cover fundamental Calculus I topics such as limits, continuity, average and instantaneous rates of change, tangent lines, and the Intermediate Value Theorem. The examples and explanations are expanded for clarity and completeness.