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  2. AP Calculus
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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)

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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.

Explain how limits at infinity relate to horizontal asymptotes.

The limit of a function as x approaches infinity (or negative infinity) gives the y-value of the horizontal asymptote, if it exists.

Describe the 'bottom heavy' rule for horizontal asymptotes of rational functions.

If the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is y = 0.

Describe the 'top heavy' rule for horizontal asymptotes of rational functions.

If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Describe the 'equal degree' rule for horizontal asymptotes of rational functions.

If the degree of the numerator equals the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients.

Explain why exponential functions grow faster than polynomial functions.

Exponential functions involve repeated multiplication, leading to much more rapid growth compared to the additive nature of polynomial functions.

Why is considering growth rates important when evaluating limits at infinity?

Growth rates help determine which part of a function dominates as x approaches infinity, simplifying the limit evaluation.

Explain why lim⁡x→∞sin⁡(x)\lim_{x \to \infty} \sin(x)limx→∞​sin(x) does not exist.

The sine function oscillates between -1 and 1 indefinitely, never approaching a single value as x goes to infinity.

Describe the growth rate hierarchy.

log < root < polynomial < exponential

What does it mean for a function to have a limit of infinity?

The function's value increases without bound as x approaches a certain value or infinity.

What is the significance of vertical asymptotes?

They indicate points where the function approaches infinity (or negative infinity) because the denominator approaches zero.

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.