Limits are played a very important role in calculus and finding the solution of complex problems by using some mathematical techniques. It is used to find the solution of the function and approximate its value at a certain point.
The idea of limits is used to explain the particular behavior of the different functions at a certain point and plays a crucial role in the development of derivative and integral calculus. Limits of different functions and complex problems are found using the different techniques of mathematics.
The direct substitution method, factoring the terms, and L’Hopitall’s rule are used to find the limits of complex problems. Limits play a very important role in the development of different fields such as engineering, solutions of physical problems in physics, and many other fields of sciences.
In this article, we will discuss the definition of limits, techniques to solve the limits, and solving examples of limits for better understanding.
Limits Definition:
Let the function F(x) and the real number “b” then limits of function F(x) is defined as,
Lim x→b F(x)= M
If “x” approaches to “b” then the limiting value of F(x) is M.
- Where, b = the point where the limiting value finds out.
- M = the limiting value of the given function at the point “b”.
If we can make the values of F(x) close to “M” by making x approximately close to “b” (but not exactly b). This means that the value of F(x) is comparably close to “M” as x approaches to “b”. The concept of limits is important for understanding continuity, derivatives, integrals, and many more concepts in calculus analysis.
Different Techniques of limits:
To solve the limits of functions we use different techniques in different situations. There are defined some techniques to solve the limits.
Direct Substitute the limit:
If the function is continuous at the limiting value then puts the limiting value directly and find the solution of the given function.
Example:
Find the limits of “7x2 – 2” by direct substitution method at “x” approaches to “-3”.
Solution:
Step 1: Consider the given value is equal to “F(x)”.
F(x) = 7x2 – 2
Step 2: write the Formula of the limit.
Lim x→b F(x)= M
Where, b = the point where the limiting value finds out, M = the limiting value of the given function at the point “b”.
Step 3:
Put the limiting value carefully in the above formula and simplify suitable mathematical operations.
F(x) = 7x2 – 2, b = -3
Lim x→-3 F(x) = Lim x→-3 (7x2 – 2)
Lim x→-3 F(x) = Lim x→-3 (7x2) – Lim x→-3 (2)
Put the limiting value carefully and simplify.
Lim x→-3 F(x) = 7(-3)2 – (2)
Lim x→-3 F(x) = 7(9) – (2)
Lim x→-3 F(x) = 63 – 2
Lim x→-3 F(x) = 61
Lim x→-3 F(x) = 61 is the solution of “7x2 – 2” at “x” approaches to “-3”.
A limit finder is a helpful source to evaluate the limit problems online with steps in a fraction of a second.
Factoring method:
In this method, the terms are given in the form of polynomial fractions “P(x) /Q(x)” and Q(x) ≠ 0. This method applies at that time if the direct substitution of the limit fails to solve and generate the “0/0” form of the given function by simply putting the value of limits. To solve this type of limiting function we use the factorization of the numerator or denominator to cancel the common term of numerator and denominator. Its formula is represented in the below form.
Lim x→b P(x)/Q(x)= K
Where Q(x) can never be equal to zero and “K” is the limiting value of the function.
Example:
Find the limit of (x2-49)/(x – 7) at the limiting value of “7”.
Solution:
Step 1: Consider the given value is equal to “F(x)”.
F(x) = (x2-49)/(x – 7)
Step 2: write the Formula of the limit.
Lim x→b F(x)= Lim x→b P(x)/Q(x)
Where Q(x) can never be equal to zero.
Step 3:
Put the limiting value carefully in the above formula and simplify suitable mathematical operations.
F(x) = (x2-49)/(x – 7), b = 7
Lim x→7 F(x) = Lim x→7 [(x2-49)/(x – 7)]
Lim x→7 F(x) = Lim x→7 (x2-49)/ Lim x→7 (x – 7)
Step 4: Put the limiting value in the above expression.
Lim x→7 F(x) = [(7)2-49] / [(7) – 7]
Lim x→7 F(x) = [49 – 49] / [7 – 7]
Lim x→7 F(x) = 0/0
We get a “0/0” form, for the solution of the given function apply the “Factoring method”.
Step 4: Factorize the term “(x2-49)” and simplify the terms.
Lim x→7 F(x) = Lim x→7 (x + 7) (x – 7)/(x – 5)
Lim x→7 F(x) = Lim x→7 (x + 7)
Step 5: Put the limiting value to get the solution of the above expression.
Lim x→7 F(x) = (7 + 7)
Lim x→7 F(x) = 14
Lim x→7 F(x) = 14 is the limiting value of (x2-49)/(x – 7) as “x” approaches to “7”.
L’Hospital’s Method:
This method uses at that time if we put the limits then we obtain the indeterminate forms such as 0/0 or ꝏ/ꝏ. To eliminate this difficulty we take the derivative of the numerator and denominator and put the limiting value and find the solution of limits. Repeat the above process at a time when we get the limiting value at a certain point.
Example:
Find the limit of the function “Sin(x)/[1-Cos2(x)]” at “x” approaches to “0”
Solution:
Step 1: Consider the given value is equal to “F(x)”.
F(x) = Sin(x)/[1- Cos2(x)]
Step 2: Apply the limiting value on both sides carefully on the above expression.
Lim x→0 F(x) = Lim x→0 {Sin(x)/[1- Cos2(x)]}
Lim x→0 F(x) = Lim x→0 Sin(x)/ Lim x→0 [1- Cos2(x)]
Step 3: Apply the limiting value.
Lim x→0 F(x) = Sin(0)/ [1- Cos2(0)]
Lim x→0 F(x) = (0)/ [1-1]
Lim x→0 F(x) = 0/0
Step 4: In the above expression we obtain an indeterminate form and apply the L’ Hospital rule.
Lim x→0 F(x) = Lim x→0 d/dx Sin(x)/ d/dx [1- Cos2(x)]
Lim x→0 F(x) = Lim x→0 Cos (x) / -2 Cos(x) (- Sin(x))
Lim x→0 F(x) = Lim x→0 Cos (x)/2 Cos(x) Sin(x)
Lim x→0 F(x) = Lim x→0 1/2 Sin(x)
Step 5: Now put the value of the limit carefully.
Lim x→0 F(x) = 1/2 Sin(0)
Lim x→0 F(x) = 1/0
Lim x→0 F(x) = ꝏ
Lim x→0 F(x) = ꝏ is the solution of “Sin(x)/[1-Cos2(x)]” at “x” approaches to “0”.
Summary:
In this article, we discussed the definition of a limit and different techniques to solve the limit in different situations. Moreover, for a better understanding of the limit concept we solved the related examples with detailed explanations.
