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  2. AP Calculus
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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.

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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 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.

How do you find the horizontal asymptote of f(x)=3x2+1x2−2f(x) = \frac{3x^2 + 1}{x^2 - 2}f(x)=x2−23x2+1​?

Compare degrees of numerator and denominator. They are equal (degree 2). Divide leading coefficients: 3/1 = 3. HA is y = 3.

How do you evaluate lim⁡x→∞xex\lim_{x \to \infty} \frac{x}{e^x}limx→∞​exx​?

Recognize exponential growth is faster than polynomial. The limit is 0.

How do you find horizontal asymptotes of rational functions?

Compare the degrees of the numerator and denominator. Apply the 'bottom heavy', 'top heavy', or 'equal degree' rules.

How do you evaluate limits involving oscillating functions at infinity?

Check if the oscillating function is divided by a function that approaches infinity. If so, the Squeeze Theorem often applies, resulting in a limit of 0. Otherwise, the limit DNE.

How do you evaluate lim⁡x→∞4x+sin⁡(x)x\lim_{x \to \infty} \frac{4x + \sin(x)}{x}limx→∞​x4x+sin(x)​?

Divide each term by x: lim⁡x→∞(4+sin⁡(x)x)\lim_{x \to \infty} (4 + \frac{\sin(x)}{x})limx→∞​(4+xsin(x)​). Since lim⁡x→∞sin⁡(x)x=0\lim_{x \to \infty} \frac{\sin(x)}{x} = 0limx→∞​xsin(x)​=0, the limit is 4.

How do you determine the end behavior of a polynomial function?

Consider the leading term of the polynomial. If the degree is even, both ends go to +∞ or -∞ (depending on the sign of the leading coefficient). If the degree is odd, one end goes to +∞ and the other to -∞.

How do you evaluate lim⁡x→∞ln⁡(x)x\lim_{x \to \infty} \frac{\ln(x)}{x}limx→∞​xln(x)​?

Recognize that polynomial growth is faster than logarithmic. The limit is 0.

How to find vertical asymptotes of a rational function?

Set the denominator equal to zero and solve for x. These x-values are potential vertical asymptotes.

How do you evaluate lim⁡x→−∞ex\lim_{x \to -\infty} e^xlimx→−∞​ex?

As x approaches negative infinity, exe^xex approaches 0.

How do you solve for horizontal asymptotes when given a function?

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