what you'll learn...
A function Defined by Limits
» A function f(x)f(x) at x=ax=a is
→ continuous: if f(a)f(a) = LHL = RHL
→ defined by value: if f(a)f(a) is a real number
→ defined by limit: if f(a)=00f(a)=00 and LHL = RHL
→ not defined: if LHL ≠ RHL and f(a)∉ℝ
function, defined at a input
A function f(x) is defined at the input value x=a, when f(a) evaluates to a defined value.
If f(a) evaluates to ∞ or 00, then the function need to be further examined.
If the left-hand-limit and right-hand-limit take a defined value and are equal, then the function is defined at that input value.
This is understood from
• In an application scenario, the value is not exact and so the function will evaluate to a stable value given by limits.
• In an abstract scenario, the expected value at that input value can be taken to be the value of the function.
A function is defined at an input value, if the function evaluates to a definite value OR if the left-hand-limit and right-hand-limits are equal.
The word 'defined' means: give exactly; state or describe without any ambiguity.
summary
Function is defined: Given a function f(x), the function is defined at x=a
•
if f(a) is a defined value. OR
• if f(a) is an indeterminate value 00 or undefined large ∞, then limx→a+f(x)=limx→a-f(x)
defined vs continuous
Difference between Defined and Continuous :
A function is defined if it evaluates to a definite value OR when is the function evaluates to indeterminate value 00, then left-hand-limit and right-hand-limit are equal.
A function is continuous only when it evaluates to a definite value at the input value AND also left-hand-limit and right-hand-limit are equal to the definite value at the input value.
• For a function to be continuous at an input value, it has to be defined at that input value.
• If a function is defined at an input value, it does not imply it is continuous. It may be or may not be continuous.
• If a function is not defined at an input value, it implies it is not continuous.
If a function is not continuous at an input value, can the function be defined at that input value?
'Yes.'. Theoretically a function can have a value f(a) different to the limits.
summary
Function is defined: Given a function f(x), the function is defined at x=a
•
if f(a) is a defined value. OR
• if f(a) is an indeterminate value 00 or undefined large ∞, then limx→a+f(x)=limx→a-f(x)
A function f(x) at x=a is
→ continuous: if f(a) = LHL = RHL
→ defined by value: if f(a) is a real number
→ defined by limit: if f(a)=00 and LHL = RHL
→ not defined: if LHL ≠ RHL and f(a)∉ℝ
Outline
The outline of material to learn "limits (calculus)" is as follows.
Note : click here for detailed outline of Limits(Calculus).
→ Indeterminate and Undefined
→ Indeterminate value in Functions
→ Expected Value
→ Continuity
→ Definition by Limits
→ Geometrical Explanation for Limits
→ Limit with Numerator and Denominator
→ Limits of Ratios - Examples
→ L'hospital Rule
→ Examining a function
→ Algebra of Limits
→ Limit of a Polynomial
→ Limit of Ratio of Zeros
→ Limit of ratio of infinities
→ limit of Binomial
→ Limit of Non-algebraic Functions