Difference Between Removable Continuity and Limit

In calculus, limits and continuity are frequently used for different purposes. The limit is a branch of calculus used to define other branches of calculus like derivative and integral. While continuity is used to check whether the function is continuous or not.

The main difference between limit and continuity is that the limit is a certain value while the continuity describes the behavior of the function. In this article, we'll cover all the basics of limit and continuity along with definitions and formulas.

What is the Limit?

In calculus, the value of the function of "x" approaches as its x-value approaches a certain value is known as a limit. In simple words, let f(x) be a function that is defined on some open interval of a number "c" except possibly at "c" itself.

Then we say that the limit of f(x) as x approaches "c" is P, and the limit can be written as:

lim _x _→ _c f(x) = P

How to calculate limits?

The problems of limits can be calculated either by using the types & rules of limits or a limit calculator. Let us take an example of a limit to learn how to calculate it.

Example 1

Find the limit of 9x³ – 19x² + 19xy + x⁵ + 2, as x approaches 7.

Solution

Step 1: Write the given function and apply the notation of limit on it.

lim_x _→ ₇ [9x³ – 19x² + 19xy + x⁵ + 2]

Step 2: Now use the sum and difference rules of limits and write the notation of limit along with each function separately.

lim_x _→ ₇ [9x³ – 19x² + 19xy + x⁵ + 2] = lim_x _→ ₇ [9x³] – lim_x _→ ₇ [19x²] + lim_x _→ ₇ [19xy] + lim_x _→ ₇ [x⁵] + lim_x _→ ₇ [2]

Step 3: Now apply the constant function rule of the limit and take the constant coefficients outside the limit notation.

lim_x _→ ₇ [9x³ – 19x² + 19xy + x⁵ + 2] = 9lim_x _→ ₇ [x³] – 19lim_x _→ ₇ [x²] + 19ylim_x _→ ₇ [x] + lim_x _→ ₇ [x⁵] + lim_x _→ ₇ [2]

Step 4: Now use the power and constant rules of limits and apply x = 7 to each function.

lim_x _→ ₇ [9x³ – 19x² + 19xy + x⁵ + 2] = 9 [7³] – 19 [7²] + 19y [7] + [7⁵] + [2]

= 9 [343] – 19 [49] + 19y [7] + [16807] + [2]

= 3087 – 931 + 133y + 16807 + 2

= 18965 + 133y

What is the Continuity?

Continuity describes the behavior of a function at a certain point or section. Limits are used to find the continuity of the function. A function f(x) is continuous at a point "c" if and only if the below conditions are satisfied.

f(c) is defined (function must be defined at "c")
lim_x _→ _c f(x) exists
lim_x _→ _c f(x) = f(c)

While if the function is not continuous, it must be a discontinuous function.

How to calculate continuity?

By satisfying the three conditions of continuity, you can determine any continuous or discontinuous functions. Let's take an example of continuity to learn how to calculate it.

Example

Determine whether f(x) = 4x² + 12x + 6 is continuous or discontinuous if x approaches 3.

Solution

Step 1: Check the given function is defined or not at x = 3

f(x) = 4x² + 12x + 6

f(3) = 4(3)² + 12(3) + 6

= 4(9) + 12(3) + 6

= 36 + 36 + 6

= 78

Hence the given function is defined at x = 3.

Step 2: Now check whether the limit of the function exists or not at x = 3.

Apply the notation of limit.

lim_x _→ ₃ f(x) = lim_x _→ ₃ [4x² + 12x + 6]

Apply the notation of limit separately to each function by using the sum rule of limits.

lim_x _→ ₃ [4x² + 12x + 6] = lim_x _→ ₃ [4x²] + lim_x _→ ₃ [12x] + lim_x _→ ₃ [6]

Now apply the constant function rule of limit and take the constant coefficients outside the limit notation.

lim_x _→ ₃ [4x² + 12x + 6] = 4lim_x _→ ₃ [x²] + 12lim_x _→ ₃ [x] + lim_x _→ ₃ [6]

Now use the power and constant rules of limits and apply x = 7 to each function.

lim_x _→ ₃ [4x² + 12x + 6] = 4 [3²] + 12 [3] + [6]

= 4 [9] + 12 [3] + [6]

= 36 + 36 + 6

= 78

Hence the limit of the function exists at x = 3.

Step 3: Now check the results of defined function and the limit.

f(3) = 78

lim_x _→ ₃ f(x) = 78

So, lim_x _→ ₃ f(x) = f(3)

Hence all the conditions of the continuity are satisfied, so the given function is continuous at x = 3.

Comparison Chart: Limit Vs Continuity

Features	Limit	Continuity
Definition	The limit is a certain value	The continuity describes the behavior of the function
Formula	f(x) = P	Holds three conditions f(c) is defined lim_x _→c f(x) exists lim_x _→c f(x) = f(c)
Application	Limits are used to define derivative, continuity, and integral	Continuity is used to check whether the function is continuous or discontinuous
Usage	Limits use the types and rules to solve their problems	Continuity uses the limits to solve its problems

Frequently Asked Questions

What is the meaning of limit and continuity?

The limit finds the numerical value of the one variable function while the continuity shows the function's behavior whether it is continuous on the given point or discontinuous.

The function must be continuous if the function is defined at a particular point, the limit of the function exists at the particular point, and the output of the limit and function must be the same at the particular point.

What are the three conditions of a continuous function?

A function is said to be continuous if the flown conditions hold.

- The function must be defined at a particular point.
- The limit of the function must exist at a specific point.
- The output of the limit and function must be identical at the specific point.

Can a limit exist if it is discontinuous?

No, the limit of the continuous function cannot exist. The limit first makes the function continuous by using the theorems of limits and then finding the limit.

What are the 4 types of discontinuity?

There are four types of discontinuity:

1 - Jump discontinuity
2 - Removable discontinuity
3 - Point discontinuity
4 - Essential discontinuity

What are the 8 limit rules?

1 - Sum rule:
lim_x→c [f(x) + h(x)] = lim_x→c [f(x)] + lim_x→c [h(x)]

2 - Constant rule
lim_x→c [K] = k

3 - Constant function rule
lim_x→c [K * f(x)] = K lim_x→c [f(x)]

4 - Power rule
lim_x→c [f(x)]ⁿ = [lim_x→c f(x)]ⁿ

5 - Difference rule:
lim_x→c [f(x) - h(x)] = lim_x→c [f(x)] - lim_x→c [h(x)]

6 - Quotient rule:
lim_x→c [f(x) / h(x)] = lim_x→c [f(x)] / lim_x→c [h(x)]

7 - Product rule:
lim_x→c [f(x) * h(x)] = lim_x→c [f(x)] * lim_x→c [h(x)]

8 - L'hopital's rule:
lim_x→c [f(x) / h(x)] = lim_x→c [d/dx f(x) / d/dx h(x)]

Conclusion

In this article, we have discussed all the basics of the limit and continuity. The limit and continuity are two different terms used to define various terms. Now you can grab all the basics of limit and continuity from the above post.

References

https://www.youtube.com/watch?v=76OovhJZozQ
https://www.youtube.com/watch?v=G0ax2x2_Em0

groesbeckfracks.blogspot.com

Source: https://www.10differences.org/difference-between-limit-and-continuity/