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Thought-Process to Discover Knowledge
Welcome to the only place where the essence of "limit of a function" is explained.
• `0/0` is called as indeterminate value -- meaning a function evaluating to `0/0` can take any value, it could be `0`, or `1`, or `7`, or `oo`, or undefined.
• other forms of indeterminate values are: `oo/oo`, `oo-oo`, `0^0`, `0xx oo`, or `oo^0`
Rigorous arithmetic calculations may result in `0/0`, but the expression may take some other value. The objective of limits is to find that value. The details explained are revolutionary and provided nowhere else.
Once that is explained, the topics in limits are covered.
(click for the list of lessons in this topic)
Basics: Limit of a function
One and only place where the essence of limit(calculus) is explained.
• Indeterminate Value
→ represented by an expression
→ other forms: `oo/oo`, `oo-oo`, `0^0`, `0xx oo`, or `oo^0`
• The following can be true
→ `0/0 = 0`
→ `0/0 = 1`
→ `0/0 = oo`
→ `0/0 = 6` or `8` or `-3`
Understanding the above is essential to understanding limits(calculus). The topic involves figuring out the expected value of a function when it evaluates to `0/0`.
Understanding limits with Graphs
Welcome to the astoundingly clear and simple lesson on understanding limits. The geometrical meaning of left-hand-limit and right-hand-limit are explained with graph of a function.
The function is considered as two constituent functions of numerator and denominator and using the graphs of these functions, the limit is explained.
Based on the understanding of numerator and denominator, the L'Hospital's Rule is explained.
Examining a function at an input value is made simple and clear. Based on the information, how to determine if the function is defined, continuous, or not defined. The following are covered.
• examining a function at an input value
• limit of a continuous function
• limit of a piecewise function
• limit of functions with abrupt change
• limit of functions that are not defined
Algebra of Limits
In this page, Algebra of limits is detailed in a coherent and simple form -- the following are covered.
• understanding algebra of limits,
• Limit distributes over Addition and Subtraction when the value is not `oo-oo`.,
• Limit distributes over multiplication, when the value is not `oo xx 0`,
• Limit distributes over division when the value is not `0 -: 0` or `oo -:oo`,
• Limit distributes over exponent, when the value is not `oo^0` or `0^0`
• Limit with a variable can be substituted when value is not any of the forms of `0/0`
Limit of Algebraic Expressions
Standard results for limits of function involving polynomials and evaluating to `0/0`, `oo/oo`, or `oo - oo` are examined and explained with examples.
• limit of polynomials
• limit of functions evaluating to `0/0`
• limit of functions evaluating to `oo/oo`
• limit of binomials
• limit of some standard trigonometric, logarithmic, and exponent functions