Limit Of A Function At A Point
Limit of a function at a point:- Limit of a function at a point indicates the behavior of a function in near a particular point or input
. If we have given a function f(x) which is a real valued function and c is a real number,
lim_(x->c)f(x) = L
It means that the f(x) is a real valued function and its value at a given point is L. We can say it in other words that the limit of the function f(x) is L at point c. The value L is in nearer and nearer to the point c.
This is the limit of a function at a point .
Evaluation of the Limit of a Function at a Point:-
Consider we have given a real valued function f(x) and we have to evaluate the limit at point a, then we write it as following way-
lim_(x->a)f(x)
since, we have to evaluate the limit at point a, so-
lim_(x->a)f(x) = f(a)
Where f(a) is obtained by putting a in place of x in the given real valued function. So the Limit of a function f(x) at a point a is f(a).
Examples of the Limit of a Function at a Point:-
1. Evaluate : lim_(x->1) 3x^2 + 4x +5.
Solution:- We have,
lim_(x->1) 3x^2 + 4x + 5.
Now, putting the limit i.e., putting 1 in place of x in the given function-
lim_(x->1) 3x^2 + 4x + 5 = 3(1)^2 + 4(1) + 5
So, lim_(x->1) 3x^2 + 4x + 5 = 3 + 4 + 5 = 12.
2. Evaluate lim_(x->2) (x^2 4)/(x + 3).
Solution:- We have,
lim_(x->2) (x^2 4)/(x + 3)
Putting the limit i.e., putting 2 in place of x in the given function:-
lim_(x->2) (x^2 4)/(x + 3) = (4 4)/(2 +3)
So, lim_(x->2) (x^2 4)/(x + 3) = (0)/(5) = 0.
3. Evaluate lim_(x->0) 9
Solution:- We have,
lim_(x->0) 9
Since, We have not given any function in x, so
lim_(x->0) 9 = 9.
Introduction to probability mass function:
A probability mass function (pmf) provides the probability that a discrete variable is accurately equal to some of the value. A probability mass function varied from a probability density function because the probability density function is described only for continuous random variables. Probability mass function (pmf) gives probability that the value of the random variable X is equal to x. Denoted by p(X=x).
Definition of Probability Mass Function:
If the range of a random variable "Y" assumes a discrete set of values y1, y2, y3, ... yn, then the function "f" defined by f(yi) = P(Y = yi) is called the "Probability Function" or "Probability Mass Function" of "X". The pmf assigns a probability [P(Y = yi)] for all the possible values [yi] of the variable.
A discrete function, f(y), is a function that describes the following properties.
f (yi) 0
The probability for the variable to carry a particular number is always a positive real number.
f (yi) = 1 [i = 1, 2, 3, ... ]
The probabilities sum of every values that the variable may carry is equal to One.
Example Problems of Probability Mass Function:
1) Toss a coin once. Let X be the number of tail that occurs. Find the probability mass function of X.
sol:
Random variable:
Number of tail Elementary Events
(X=0) (Tail)
(X=1) (Tail)
Probability mass function:
X f(x)
0
1
2) Toss a coin twice. Let X be the number of tails. Find the probability mass function of X.
solution:
Denote (Head, Head) to be the elementary event that the first toss is tail and the second toss is Head. Denote the other elementary events accordingly.
Random variable:
Number of Tails (y) Elementary Events
0 (Head, Head)
1 (Head, Tail)(Tail, Head)
2 (Tail, Tail)
probability mass function:
x f(x)
0
1
2
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