what is a limit?
When a function, f(x) approaches an x value that it never actually touches, it is called a limit. The rule goes as follows: If f(x) becomes close to a number (L) as x is approaching a given value c from the left and right sides, the limit of f(x) as x approaches c is equal to the number “L” (just like an asymptote).
For this, we write:
For this, we write:
There are moments when Limits DNE (aka Do Not Exist).
When a limit DNE:
(#– or #^- = left-hand limit, from the left of # ; #+ or #^+ = right-hand limit, from the right of #)
When a limit DNE:
(#– or #^- = left-hand limit, from the left of # ; #+ or #^+ = right-hand limit, from the right of #)
- When a function increases or decreases infinitely (without bound)
lim f(x) = DNE ; lim f(x) = DNE
x –> ∞ +∞ x –> -∞ +∞
- When both left and right-hand limits are not equal to one another
lim g(x) = 0 ; lim g(x) = 3 ; lim g(x) = DNE since Left and Right limits are not equal
x –> -2– x –> -2+ -2
- When the function oscillates between two values
lim h(x) = DNE
x –> ∞
Limits can be evaluated through graphing, direct substitution, inspection and other algebraic methods.
• Graphing
Analyzing graph around x = c
• Direct Substitution
Substituting the value of x = c into function and evaluating (finding f(c) which represents the limit).
• Inspection
Like mental math, talking/reasoning to understand what is happening to the y-values as the x-values approach c.
• Algebraically
Manipulating the function algebraically
• Graphing
Analyzing graph around x = c
• Direct Substitution
Substituting the value of x = c into function and evaluating (finding f(c) which represents the limit).
• Inspection
Like mental math, talking/reasoning to understand what is happening to the y-values as the x-values approach c.
• Algebraically
Manipulating the function algebraically
Limit laws
laws of limits are held true supposing that b and c are some real numbers, n is some positive integer, and that the lim f(x) as x –> a and the lim g(x) as x –> a both exist.
methods: limits as "x" approaches infinity
When direct substitution does not work for approaching infinity, aka running into "∞/∞" or "0/0", you must do other things to evaluate.
Cases: dividing by highest power of "x" and approaching infinity
Case I
When the numerator is
smaller than
the denominator,
the limit is always
equal to 0.
When the numerator is
smaller than
the denominator,
the limit is always
equal to 0.
Case 2
When the powers are equal,
the limit is equal to the coefficient of the numerator divided by the coefficient
of the denominator.
When the powers are equal,
the limit is equal to the coefficient of the numerator divided by the coefficient
of the denominator.
Case 3
When the power of the numerator is larger than
the power of the
denominator, the
limit does not exist.
When the power of the numerator is larger than
the power of the
denominator, the
limit does not exist.
NOTE: Not all limits involving ∞ and 0 can be evaluated. See what this is all about here.