Limit Basics
What does the notation “limₓ→a f(x)” represent?
The value f(x) approaches as x gets closer to a.
Find limₓ→3 (x² - 9) / (x - 3).
6
What sign represents the left-hand limit at x = a?
limₓ→a⁻ f(x)
Minus sign
What does limₓ→a f(x) = ∞ mean?
f(x) increases without bound as x approaches a.
True or False: A limit always equals the function’s value at that point.
False
What algebraic technique removes a 0/0 indeterminate form from a rational expression?
Factoring and canceling common terms.
What symbol represents the right-hand limit at x = a?
limₓ→a⁺ f(x)
Plus sign
If a function grows without bound near x = 3, what kind of asymptote occurs?
A vertical asymptote at x = 3.
What must be true for a limit to exist at x = a?
The left-hand limit and right-hand limit must both exist and be equal.
Find limₓ→2 (x3 + x - 6) / (x + 2).
1
If limₓ→2⁻ f(x) = 3 and limₓ→2⁺ f(x) = 5, what is limₓ→2 f(x)?
It does not exist.
Find limₓ→2⁻ 1 / (x - 2).
−∞
If limₓ→2 f(x) = 5, what does this mean about f(2)?
f(2) could be 5 or undefined.
Find limₓ→30 (sin x) / x.
1/60
How can a removable discontinuity be shown on a graph?
A hole (open circle) where the function is undefined but the limit exists.
Find limₓ→∞ (3x² + x) / (2x² + 5).
3/2
Name 3 types of discontinuities.
Jump Discontinuity, Infinite Discontinuity, Removable Discontinuity.
Simplify and find limₓ→0 (√(x + 4) - 2) / x.
1/4
If a graph has a jump at x = a, what does that mean for the limit at a?
If a graph has a jump at x = a, what does that mean for the limit at a?
What is the horizontal asymptote of f(x) = (5x² + 2) / (x² + 7)?
y = 5