Problems on Algebraic Properties of Limits | Hindi | Maths

Limits: You and your friends decide to meet at some place outside. Is it necessary that all your friends are living in the same place and walk on the same road to reach that place? No, not always. All friends come from different parts of the city or country to meet at that one single place.

It looks like a convergence of different elements to a single point. Mathematically, it is like a convergence of a function to a particular value. It is an example of limits. Limits show how some functions are bounded. The function tends to some value when its limit approaches some value.
Introduction to Limits
Suppose we have a function f(x). The value, a function attains, as the variable x approaches a particular value say a, i.e., x → a is called its limit. Here, ‘a’ is some pre-assigned value. It is denoted as

limx→af(x) = l

The expected value of the function shown by the points to the left of a point ‘a’ is the left-hand limit of the function at that point. It is denoted as limx→a− f(x).
The points to the right of a point ‘a’ which shows the value of the function is the right-hand limit of the function at that point. It is denoted as limx→a+ f(x).
Limits of functions at a point are the common and coincidence value of the left and right-hand limits.
Properties of Limits
limx→a c = c, where c is a constant quantity.
The value of limx→a x = a
Value of limx→a bx + c = ba + c
limx→a xn = an, if n is a positive integer.
Value of limx→0+ 1/xr = +∞.
limx→0− 1/xr = −∞, if r is odd, and
limx→0− 1/xr = +∞, if r is even.
Algebra of Limits
Let p and q be two functions such that their limits limx→a p(x) and limx→a q(x) exist.

Limit of the sum of two functions is the sum of the limits of the functions.
limx→a [p(x) + q(x)] = limx→a p(x) + limx→a q(x).

Limit of the difference of two functions is the difference of the limits of the functions.
limx→a [p(x) − q(x)] = limx→a p(x) − limx→a q(x).

Limit of product of two functions is the product of the limits of the functions.
limx→a [p(x) × q(x)] = [limx→a p(x)] × [limx→a q(x)].

Limit of quotient of two functions is the quotient of the limits of the functions.
limx→a [p(x) ÷ q(x)] = [limx→a p(x)] ÷ [limx→a q(x)].

Limit of product of a function p(x) with a constant, q(x) = α is α times the limit of p(x).
limx→a [α.p(x))] = α. limx→a p(x).

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