Definition
If f(x) becomes close to a single number L as x approaches c from both the left and the right, then the limit of f(x) as x approaches c is L. This is denoted as:
lim f(x) = L
x->c
Notation
lim is the notation used.
Limit of a function
The limit of a function is the y-value that the function approaches when it is coming towards an x-value from both the left and the right.
Properties of Limits
- lim k = k as x approaches c
- lim [ f(x) + g(x)] = lim f(x) + lim g(x)
- lim f(x) - g(x) = lim f(x) - lim g(x)
Special Trig Limits
- lim sin x/x = 1 as x approaches 0
- lim sin x = 0 as x approaches 0
- lim sin 2x/x = 2 as x approaches 0
Graphically
Limit can be find graphically by observing the graph.
Algebraically
- Method A: Substitution
- Method B: Factor & Cancel
- Method C: Use special trig limits
- Method D: Logic
Involving Infinity
Limits can be calculated involving infinity using specific math operations and properties.
Horizontal Asymptotes
Property: lim f(x) = c as x approaches ±∞
Vertical Asymptotes
Property: lim f(x) = ±∞ as x approaches a value that makes the denominator equal to 0.
Definition & Implications
Continuous functions can be drawn in one stroke without lifting your pen, i.e., at interior points within a domain. The continuity is defined by the three conditions:
1) f(c) must be defined
2) lim f(x) must exist
3) f(c) = lim f(x)
Discontinuity
Discontinuity is the opposite of continuity. It means the function cannot be drawn with one stroke.
Intermediate Value Theorem
The theorem states that if a function is continuous on [a, b] and K is any number between f(a) and f(b), then there is at least one c in [a, b] such that f(c) = k.
- Find the value of a that will make f(x) = x²+4 = 5ax
- Determine if g(-3) is defined
- Find if lim x+3 g(x) exists
- Check if g(-3) equals lim x+3 g(x)
Average vs. Instantaneous ROC
Example:
Find the slope of the curve at x=-3 for the function f(x) = x².
By understanding the properties and definitions of limits, continuity, and slopes of curves, these concepts can be applied and practiced in solving problems and understanding calculus. The correct application of these concepts is crucial in mastering Calculus AB.
For further detailed understanding and practice of these concepts, refer to AP Calc AB concepts, Limits notes for ap calc ab pdf, ap calculus limits practice pdf, ap calculus: limits test with answers, ap calculus limits and continuity test pdf, ap calculus ab limits and continuity practice answers, ap calculus ab notes pdf, ap calculus ab limits questions, ap calc ab limits and continuity, properties of limits pdf, Properties of limits calculator, properties of limits examples, properties of limits proof, properties of limits of trigonometric functions, limit laws, properties of limits worksheet, properties of limits composite functions, Special trig limits proof, Special trig limits examples, Special trig limits calculator, special trig limits worksheet with answers, special trig limits definition, limits of trigonometric functions examples and solutions pdf.
