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

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 $\lim_{x \to \infty} \sin(x)$ 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?

$\lim_{x \to \infty} f(x)$

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

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

If degree(p(x)) < degree(q(x)), what is $\lim_{x \to \infty} \frac{p(x)}{q(x)}$ ?

If degree(p(x)) = degree(q(x)), what is $\lim_{x \to \infty} \frac{p(x)}{q(x)}$ ?

Ratio of the leading coefficients.

What is $\lim_{x \to \infty} \frac{\sin(x)}{x}$ ?

General form of a rational function.

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

What is the horizontal asymptote if $\lim_{x \to \infty} f(x) = L$ ?

y = L

What is the general form of an exponential function?

$f(x) = a^x$ , where a is a constant.

What is the general form of a logarithmic function?

$f(x) = \log_a(x)$ , 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.