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  2. AP Calculus
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Define 'limit'.

The value that a function approaches as the input approaches some value.

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Define 'limit'.

The value that a function approaches as the input approaches some value.

What is a horizontal asymptote?

A y-value that the graph of a function approaches as x tends to +∞ or -∞.

Define 'end behavior'.

The behavior of a function as x approaches positive or negative infinity.

What is an oscillating function?

A function that repeatedly fluctuates between two or more values as x approaches infinity.

Define the Squeeze Theorem.

If g(x) ≤ f(x) ≤ h(x) for all x near c, and lim x→c g(x) = L = lim x→c h(x), then lim x→c f(x) = L.

What is meant by 'growth rate' in the context of functions?

The speed at which a function's output increases as its input increases.

Define a rational function.

A function that can be defined as a fraction where both the numerator and denominator are polynomials.

What is the degree of a polynomial?

The highest power of the variable in the polynomial.

Define 'leading coefficient'.

The coefficient of the term with the highest power in a polynomial.

What does DNE stand for in the context of limits?

Does Not Exist

What are the differences between limits at infinity and limits at a finite number?

Limits at infinity: x approaches ∞ or -∞, examining end behavior. Limits at a finite number: x approaches a specific value, examining local behavior.

What are the differences between horizontal and vertical asymptotes?

Horizontal: y-value function approaches as x approaches infinity. Vertical: x-value function approaches infinity.

Compare the growth rates of polynomial and exponential functions.

Polynomial: Grows at a polynomial rate (e.g., x2x^2x2, x3x^3x3). Exponential: Grows at an exponential rate (e.g., 2x2^x2x, exe^xex). Exponential growth is much faster.

Compare the behavior of 1x\frac{1}{x}x1​ and 1x2\frac{1}{x^2}x21​ as x approaches infinity.

Both approach 0. 1x\frac{1}{x}x1​ can be positive or negative depending on the sign of x, while 1x2\frac{1}{x^2}x21​ is always positive.

Contrast the limits of sin⁡(x)\sin(x)sin(x) and sin⁡(x)x\frac{\sin(x)}{x}xsin(x)​ as x approaches infinity.

sin⁡(x)\sin(x)sin(x): Limit DNE (oscillates). sin⁡(x)x\frac{\sin(x)}{x}xsin(x)​: Limit is 0 (Squeeze Theorem).

Compare the end behavior of x2x^2x2 and x3x^3x3.

x2x^2x2: As x approaches ±∞, x2x^2x2 approaches +∞. x3x^3x3: As x approaches +∞, x3x^3x3 approaches +∞. As x approaches -∞, x3x^3x3 approaches -∞.

Compare horizontal asymptotes with removable discontinuities.

Horizontal Asymptotes: Describe the end behavior of a function. Removable Discontinuities: Points where the function is not defined but could be redefined to be continuous.

Compare the limits of ln⁡(x)\ln(x)ln(x) and exe^xex as x approaches infinity.

ln⁡(x)\ln(x)ln(x): Approaches infinity, but very slowly. exe^xex: Approaches infinity very quickly.

What is the difference between an infinite limit and a limit at infinity?

Infinite Limit: The function approaches infinity as x approaches a finite value. Limit at Infinity: The function approaches a finite value or infinity as x approaches infinity.

Compare the limits of x+1x\frac{x+1}{x}xx+1​ and xx+1\frac{x}{x+1}x+1x​ as x approaches infinity.

x+1x\frac{x+1}{x}xx+1​: Approaches 1. xx+1\frac{x}{x+1}x+1x​: Approaches 1.

What is the notation for the limit of f(x) as x approaches infinity?

lim⁡x→∞f(x)\lim_{x \to \infty} f(x)limx→∞​f(x)

What is the notation for the limit of f(x) as x approaches negative infinity?

lim⁡x→−∞f(x)\lim_{x \to -\infty} f(x)limx→−∞​f(x)

If degree(p(x)) < degree(q(x)), what is lim⁡x→∞p(x)q(x)\lim_{x \to \infty} \frac{p(x)}{q(x)}limx→∞​q(x)p(x)​?

0

If degree(p(x)) = degree(q(x)), what is lim⁡x→∞p(x)q(x)\lim_{x \to \infty} \frac{p(x)}{q(x)}limx→∞​q(x)p(x)​?

Ratio of the leading coefficients.

What is lim⁡x→∞sin⁡(x)x\lim_{x \to \infty} \frac{\sin(x)}{x}limx→∞​xsin(x)​?

0

General form of a rational function.

p(x)q(x)\frac{p(x)}{q(x)}q(x)p(x)​ where p(x) and q(x) are polynomials

What is the horizontal asymptote if lim⁡x→∞f(x)=L\lim_{x \to \infty} f(x) = Llimx→∞​f(x)=L?

y = L

What is the general form of an exponential function?

f(x)=axf(x) = a^xf(x)=ax, where a is a constant.

What is the general form of a logarithmic function?

f(x)=log⁡a(x)f(x) = \log_a(x)f(x)=loga​(x), where a is a constant.

What does the notation ∞∞\frac{\infty}{\infty}∞∞​ suggest when evaluating limits?

Consider using horizontal asymptote rules or L'Hopital's Rule.