In mathematics, limit and continuity are most widely and frequently used in calculus. Continuity is a method to perform a task in a continuous way.
As once you start to trace the pencil to draw a graph of a function, if you do the tracing without lifting the pencil then that function must be a continuous function. While on the other hand limit is a function that approaches a specific value.
Limits are very useful in calculus for the calculations of integral, continuity, and derivatives. They are very helpful for solving the problems related to integral, continuity, and derivatives. Limits and continuity are combinedly used in a large number of ways in calculus and mathematics.
Limits – Definition and Introduction
In calculus, a function that approaches the final result for the given initial values is said to be the limit of that function. Limits are used in calculus to identify the continuity, derivative, and integral. The main purpose of the limit is to define these above-mentioned methods. Limits are widely used to tell the behavior of the function or to analyze the process of the function at a particular point.
Limits are also used in the theory category and the topological net for further generalization. Limits are generally very helpful in the integrals. As integral are classified into two ways one is the definite integral and the other is the indefinite integral. For the calculation of the definite integral, we must have the upper and lower limits of the given function. On the other hand, the indefinite integral is calculated without limits, just by following the basic formulas.
Limits are also helpful for the calculations of derivatives by the first principal. In the first principle of the derivatives, limits are used to perform the calculations and give the final result.
Limits are generally, defined as a function approach to the given values, as x approaches n give the output M,
= M
Limits are used to calculate the perfect result of the given input. Limit calculator can be used to calculate the accurate results of limits such as left-hand limit, right-hand limit, or two-sided limit. All the results will be precise and you will also get the whole solution with steps.
Types of Limits
There are three main types of the limit. These types are frequently used in calculus for the representation of different scenarios of the given function.
- Left-hand Limit
The first type of limit is the left-hand limit also written as LHL, which is a limit in with function that behaves to the given input at the left as immediate left form x=n. can be expressed as
= M
- Right-hand Limit
The second type of limit is the right-hand limit also written as RHL, which is a limit with a function that behaves to the given input at the right as immediate right form x=n. can be expressed as
= M
- Two-sided Limit
When the limits give the precise value of the given function at x=n. in two-sided limit left-hand limit is equal to the left-hand limit and is written as,
= M
How to calculate limit problems?
For the calculation of limits, let’s use some examples.
Example 1
Evaluate 7x + 8
Solution
Step 1: Apply the sum rule.
7x + 8 = 7x + 8
Step 2: Apply constant rule.
7x + 8 = + 8
Step 3: Apply limits.
7x + 8 = 7(3) + 8
= 21 + 8 = 29
Example 2
Evaluate x2 + 6x + 9 / x2 – 9
Solution
Step 1: Factorize the numerator.
x2 + 6x + 9 / x2 – 9 = (x + 3) (x + 3) / x2 – 9
Step 2: Factorize the denominator.
x2 + 6x + 9 / x2 – 9 = (x + 3) (x + 3) / (x + 3) (x – 3)
Step 3: Cancel the common numerator and denominator.
= (x + 3) (x + 3) / (x + 3) (x – 3)
= (x + 3) / (x – 3)
Step 4: Apply the quotient rule.
= (x + 3) / (x – 3)
Step 5: Apply limits:
= (4 + 4) / (4 – 3)
= 8 /1 = 8
Step 4: Write the equation with the result.
x2 + 6x + 9 / x2 – 9 = 8
Continuity
Continuity is a method to perform a task in a continuous way. As once you start to trace the pencil to draw a graph of a function, if you do the tracing without lifting the pencil then that function must be a continuous function.
In calculus, continuity can be written mathematically by the following steps.
- Given function must be defined at x=n as f(x) = f(n)
- There must be exist left hand limit (f(x)) and right-hand limit ().
- When the left-hand limit becomes equal to the right-hand limit it must be equal to the continuity.
= = f(n)
When all the above steps are satisfied on the given function then the given function is said to be the continuous function.
Relation between Limit and Continuity?
When the value of f near x to the left of a, i.e., the left-hand limit of f at a, and the value of f near x to the right of a, i.e., the right-hand limit of f at a are identical, we name that common value the limit of f(x) at x = a. Also, if the limit of f(x) as x approaches a is equal to f(a), the function f is said to be continuous at a.
The best relation between limit and continuity is that we can define continuity only with the help of limits such as “A function f with variable x is continuous at the point “a” on the real line if the limit of f(x), as x approaches the point “a,” is equal to the value of f(x) at “a,” which means f(x) is continuous f(a).
Conclusion
Limits and continuity are used in calculus mostly either to calculate the function or to define them. Limits and continuity are not much difficult topics. Once the basic concept of these topics is acquired you can easily solve any problem related to them.