Calculus Study Notes: Limits, Asymptotes, Piecewise Functions, and Tangents

Limits and Asymptotes

Vertical and Horizontal Asymptotes

Asymptotes are lines that a graph approaches but never touches. They are important in understanding the behavior of rational functions as the input grows large or approaches certain critical values.

• Vertical Asymptote: Occurs where the denominator of a rational function is zero and the numerator is nonzero.

• Horizontal Asymptote: Describes the end behavior of a function as $x$ approaches infinity or negative infinity.

• Example: For $h(x) = \frac{3x-3}{x^2-2}$:

  • Vertical asymptotes: Set denominator to zero: $x^2-2=0 \implies x=\sqrt{2}, x=-\sqrt{2}$

  • Horizontal asymptote: Compare degrees of numerator and denominator. Since degree of denominator (2) is greater than numerator (1), horizontal asymptote is $y=0$.

Piecewise Defined Functions

Definition and Graphing

A piecewise function is defined by different expressions depending on the input value. Understanding how to graph and analyze these functions is essential in calculus.

• Example:

  • $h(x) = \begin{cases} -\sqrt{x} & x < 0 \\ 3-x & 0 \leq x < 3 \\ (x-3)^2 & x \geq 3 \end{cases}$

• To graph, plot each piece on its respective domain and check for continuity at the boundaries.

Limits of Piecewise Functions

Limits describe the behavior of a function as the input approaches a specific value. For piecewise functions, check the left and right limits at points where the formula changes.

• Key Points:

  • $\lim_{x \to 0^-} h(x)$: Use the first piece ($-\sqrt{x}$).

  • $\lim_{x \to 0^+} h(x)$: Use the second piece ($3-x$).

  • $\lim_{x \to 3^-} h(x)$: Use the second piece ($3-x$).

  • $\lim_{x \to 3^+} h(x)$: Use the third piece ($(x-3)^2$).

  • $\lim_{x \to -\infty} h(x)$ and $\lim_{x \to \infty} h(x)$: Use the end behavior of the relevant pieces.

• Discontinuity: A function is discontinuous at $x=a$ if the left and right limits at $a$ are not equal or the function is not defined at $a$.

Velocity and Free Fall Problems

Application of Derivatives to Motion

Calculus is used to analyze the motion of objects, such as a ball thrown upward. The position function gives height as a function of time, and its derivative gives velocity.

• Key Formulas:

  • Position: $h(t) = -16t^2 + v_0 t + h_0$ (for vertical motion under gravity, in feet)

  • Velocity: $v(t) = h'(t) = -32t + v_0$

• Example Applications:

  • Find velocity at a specific time by substituting $t$ into $v(t)$.

  • Find when the ball hits the ground by solving $h(t) = 0$.

  • Impact velocity is the velocity when the ball reaches the ground.

Limits and Tangent Lines

Limit Definition of the Derivative

The derivative of a function at a point gives the slope of the tangent line at that point. The limit definition is fundamental in calculus.

• Definition: The derivative of $f(x)$ at $x=a$ is:

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

• Example: For $g(x) = \frac{3}{x}$, find $g'(x)$ using the limit definition.

• Tangent Line Equation: The tangent line at $(a, f(a))$ is:

  • $y = f'(a)(x - a) + f(a)$

Constructing Functions with Given Properties

Function Construction and Limits

Sometimes, you are asked to construct a function that meets specific conditions, such as values at certain points and limits at infinity.

• Example: Sketch a function $f(x)$ such that:

  • $f(0) = 0$

  • $f(0) = 3$

  • $f(3) = 2$

  • $\lim_{x \to 4} f(x) = 4$

  • $\lim_{x \to 5} f(x) = 5$

  • $\lim_{x \to -\infty} f(x) = \infty$

  • $\lim_{x \to \infty} f(x) = 0$

• Such problems test your understanding of function behavior and limits.

Summary Table: Types of Asymptotes

Type	How to Find	Example
Vertical Asymptote	Set denominator to zero	$x^2-2=0 \implies x=\pm\sqrt{2}$
Horizontal Asymptote	Compare degrees of numerator and denominator	Degree denominator > numerator: $y=0$

Additional info:

• Piecewise functions may be discontinuous at the points where the formula changes. Always check left and right limits.

• When constructing functions with given properties, you may need to use different pieces or adjust the function's formula to meet all conditions.