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 $\lim_{x \to \infty} \frac{p(x)}{q(x)}$?
0
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 $\lim_{x \to \infty} \frac{\sin(x)}{x}$?
0
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.
If a graph approaches y = 2 as x goes to infinity, what does this mean?
The function has a horizontal asymptote at y = 2, and $\lim_{x \to \infty} f(x) = 2$.
If a graph has a horizontal asymptote at y = 0, what does this imply about the function's behavior at infinity?
The function's y-values approach 0 as x goes to positive or negative infinity.
How can you identify a horizontal asymptote on a graph?
Look for a horizontal line that the graph approaches but does not cross (at least at the extreme ends).
How can you identify a vertical asymptote on a graph?
Look for a vertical line where the function approaches infinity or negative infinity.
What does a graph with no horizontal asymptote indicate?
The function's y-values do not approach a finite limit as x goes to positive or negative infinity.
What does the graph of f(x) = e^x look like as x approaches negative infinity?
The graph approaches the x-axis (y=0) from above.
What does the graph of f(x) = ln(x) look like as x approaches infinity?
The graph increases without bound, but at a decreasing rate.
How to interpret the graph of $\frac{\sin(x)}{x}$ as x approaches infinity?
The graph oscillates with decreasing amplitude, approaching y = 0.
What does it mean if a graph crosses its horizontal asymptote?
The function's value equals the value of the horizontal asymptote at that point. This is possible, but the function must still approach the asymptote as x goes to infinity.
What can you infer if a function has a horizontal asymptote at y=L?
$\lim_{x \to \infty} f(x) = L$ and $\lim_{x \to -\infty} f(x) = L$
How do you find the horizontal asymptote of $f(x) = \frac{3x^2 + 1}{x^2 - 2}$?
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 $\lim_{x \to \infty} \frac{x}{e^x}$?
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 $\lim_{x \to \infty} \frac{4x + \sin(x)}{x}$?
Divide each term by x: $\lim_{x \to \infty} (4 + \frac{\sin(x)}{x})$. Since $\lim_{x \to \infty} \frac{\sin(x)}{x} = 0$, 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 $\lim_{x \to \infty} \frac{\ln(x)}{x}$?
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 $\lim_{x \to -\infty} e^x$?
As x approaches negative infinity, $e^x$ approaches 0.
How do you solve for horizontal asymptotes when given a function?
Solve for $\lim_{x \to \infty} f(x)$ and $\lim_{x \to -\infty} f(x)$.
