Calculus I Exam Study Guide: Limits, Continuity, and Tangents

Limits and Continuity

Introduction to Limits

Limits are a foundational concept in calculus, describing the behavior of functions as inputs approach specific values. Understanding limits is essential for defining derivatives, continuity, and analyzing function behavior near points of interest.

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

• Notation: $\lim_{x \to a} f(x)$

• Key Properties:

  • If the left-hand and right-hand limits are equal, the limit exists.

  • Limits can be finite or infinite, and may not exist if the function behaves erratically near the point.

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

Evaluating Limits

Limits can be evaluated using direct substitution, factoring, rationalization, or special limit laws. Some limits require recognizing indeterminate forms and applying algebraic techniques.

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

• Factoring: Used when substitution yields 0/0; factor numerator and denominator to simplify.

• Rationalization: Multiply by a conjugate to simplify expressions involving roots.

• Special Trigonometric Limits: $\lim_{x \to 0} \frac{\sin x}{x} = 1$ and $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$.

• Example: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$

  • Factor numerator: $(x - 2)(x + 2)$

  • Simplify: $\lim_{x \to 2} x + 2 = 4$

Limits Involving Trigonometric Functions

Trigonometric limits often require using identities or recognizing standard forms.

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

  • Evaluate: $\sin(\frac{\pi}{2}) = 1$, $\cos(\pi) = -1$

  • Result: $1 - 3(-1) = 1 + 3 = 4$

• Example: $\lim_{x \to 0} \frac{\sin(5x)}{x}$

  • Use: $\lim_{x \to 0} \frac{\sin(ax)}{x} = a$

  • Result: $5$

Limits Involving Roots and Rational Functions

Limits with roots or rational expressions may require algebraic manipulation.

• Example: $\lim_{x \to 4} \frac{\sqrt{x + 2} - 3}{x - 4}$

  • Direct substitution yields 0/0; rationalize numerator.

  • Multiply numerator and denominator by $\sqrt{x + 2} + 3$.

  • Simplify and evaluate the limit.

Continuity

A function is continuous at a point if its limit exists at that point and equals the function's value there. Continuity is crucial for applying calculus concepts like differentiation and integration.

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

• Interval Notation: Used to describe where a function is continuous.

• Types of Discontinuity:

  • Removable: The limit exists, but f(a) is undefined or not equal to the limit.

  • Non-removable: The limit does not exist due to jump or infinite discontinuity.

• Example: $f(x) = \frac{x^2 - 1}{x - 1}$ is discontinuous at x = 1, but the discontinuity is removable.

Average and Instantaneous Rate of Change

Average Speed

The average rate of change of a function over an interval is the change in output divided by the change in input.

• Formula: $\text{Average Rate} = \frac{f(b) - f(a)}{b - a}$

• Application: For a skydiver with position $h(t) = -16t^2 + 2880$, average speed between $t = 0$ and $t = 60$ is $\frac{h(60) - h(0)}{60 - 0}$.

Instantaneous Speed (Derivative)

The instantaneous rate of change at a point is the derivative, found using the limit of the difference quotient.

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

• Application: For $h(t)$, the instantaneous speed at $t = 1$ is $h'(1)$.

Tangent Line to a Curve

The tangent line to a function at a point has a slope equal to the derivative at that point and passes through the point.

• Equation: $y - f(a) = f'(a)(x - a)$

• Example: For $h(t)$ at $t = 1$, the tangent line is $y - h(1) = h'(1)(x - 1)$.

Solving Equations and the Intermediate Value Theorem

Intermediate Value Theorem (IVT)

The IVT states that if a function is continuous on [a, b] and takes values f(a) and f(b), then it takes every value between f(a) and f(b) at some point in (a, b).

• Application: To show that $f(x) = x^3 - 5x^2 - 10x^2 + 15x + 30$ has a root in [2, 3], check that $f(2)$ and $f(3)$ have opposite signs.

Summary Table: Types of Discontinuity

Type	Description	Removable?	Example
Removable	Limit exists, function value missing or mismatched	Yes	$f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$
Jump	Left and right limits differ	No	Piecewise function with different values at a point
Infinite	Function approaches infinity	No	$f(x) = \frac{1}{x}$ at $x = 0$

Key Formulas and Theorems

• Limit Definition of Derivative: $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

• Average Rate of Change: $\frac{f(b) - f(a)}{b - a}$

• Equation of Tangent Line: $y - f(a) = f'(a)(x - a)$

• Intermediate Value Theorem: If $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then there exists $c \in (a, b)$ such that $f(c) = N$.

Practice Problems (from Exam)

• Compute limits involving trigonometric, rational, and root functions.

• Determine intervals of continuity and classify discontinuities.

• Apply the IVT to show existence of roots.

• Find average and instantaneous rates of change for position functions.

• Write equations of tangent lines at specified points.

Additional info: The above guide expands on the exam questions, providing definitions, formulas, and context for each topic covered. Students should practice evaluating limits, identifying continuity, and applying theorems as shown.