Today's lesson: Limits and Continuity
The limits and continuity constitutes an important part of the calculus math. It serves as a basis for many of the derivative formulas and has other important applications. So let us define the limit and go further with the topic.
Defining limit and understanding its properties:
Consider a function g(x) being defined on some interval with x = c. Then limit of a function L is given by the value that a function reaches when x pursues c. It can be formulated as
A function g(x) can be said continuous at x = c only if the limit equals g of c. It can be formulated as
Thus limits continuity can easily be found out if we know the approaching input value. Some limit and continuity properties are listed below which will help solving the examples.
The function is continuous if and only if it is defined for every points on a given interval.
The left hand limit signifies that x approaches the input from left side while in right hand limit, the value is approached from right side.
Limit is said to be unique if g(x) has that limit at x = c.
The limit of two functions for same approaching value is equal to the individual sum of limits.
Above rule holds true for subtraction, division as well as multiplication of limits.
The limit of product of a constant and the function is equal to the product of constant multiplied by limit of a function.
Thus we have seen important features of calculus limits and continuity that has made the task of solving limits and continuity examples easier.
(1) Find the limit of x2 if x approaches 3. Also check the continuity:
Since x approaches 3, we have limit L = [x2]x->3 = 32 = 9. Thus limit of square of x at x->3 is 9. Since function is defined only for a particular value of x, we can say it is continuous.
(2) Find the points where the function g(x) = 1/(x2-2x+15) is not continuous:
If the denominator turns out to be zero, we get the discontinuity for rational functions. So lets equate it with zero i.e. x2 - 2x + 15 = 0 or (x + 3)(x – 5) = 0. Hence g(x) is discontinuous at x = -3 and x = 5.
|More topics in Limits and Continuity|
|Central Limit Theorem||Limits of Trig Functions|
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