An Introduction to Limit Calculus with its Rules, Applications and Examples
An Introduction to Limit Calculus with its Rules, Applications and Examples
The limit is the approximate value of a function as its input approaches a specific number. Limit is the principle concept and core dimension of calculus. All the fundamental notions like continuous functions, derivatives, integrals, and convergent series are limits in one sense or another.
The concept of limits plays a crucial role to apprehend the geometric aspects of curves and surfaces such as length, area, slope and volume. The features of motion like speed, velocity, acceleration and momentum are best comprehended as limits as well.
Definition of Limit Calculus
If f(x) approaches a real number L as x gets closer to (but not equal to) a real value a (for both x<a and x>a), then we say that the limit of f(x) is L as x approaches a. Mathematically,
limxàa f(x) = L
We pronounce this notation as “the limit of f(X), as x approaches an equal to L”. Here x is the independent variable and a is the number to which x approaches. For the limit of a function to exist as x approaches a, it is necessary that the value of the limit of the function is the same as x approaches a from the left and right-hand sides.
Rules of the limit:
Some of the important rules of limits are given below which are used to compute the limit of functions.
Sum Rule:
If limxàa f(x) = L and limxàa g(x) = M. Then
limxàa (f(x) + g(x)) = limxàa f(x) + limxàa g(x) = L + M
Difference Rule:
limxàa (f(x) – g(x)) = limxàa f(X) – limxàa g(x) = L – M
Product Rule:
limxàa (f(x) * g(x)) = limxàa f(x) * limxàa g(x) = L * M
Quotient Rule:
limxàa (f(x) / g(x)) = limxàa f(x) / limxàa g(x) = L / M
Constant Rule:
limxàa c = c
Where c is a constant function.
L’Hospital Rule:
L’Hospital rule is one of the most important rule for the computation of derivatives and limits of the function. This rule states that if function gives 0/0 or ∞/∞ form by putting the limit value in the function, then differentiate both numerator and denominator first. After that compute the limit of the function.
This process of applying the L’Hospital rule will be repeated until the function stops over giving 0/0 or ∞/∞ form.
limxàa (f(x) / g(x)) = limxàa [f’(x) / g’(x)]
Constant Function Rule:
limxàa C f(x) = C * limxàa f(x) = C * L
Where C is a constant number.
Power Rule:
limxàa [f(x)]n = Ln, where n is a positive number.
Applications of Limits:
Limits provide a powerful tool for the mathematical analysis of objects. The concept of limits is used over a wide area of continuity, derivatives, integration (definite integration) etc. Here we will describe concisely the applications of limits.
To Compute Derivatives:
The concept of limits is applied to the computation of derivatives. For example, if we aspire to compute the slope of the tangent. Derivative concerns the points on a curve. The concept of limits helps us to consider the distance b/w the two points which is a small rate of change in the derivative context. To find out the slop of these points, we obtain the given formula
Slop = (y2 – y1) / (x2 – x1) = (f(x + Δx) – f(x)) / ((x + Δ) – Δx )
Here Δ is very small nearly equal to zero but not equal to zero. In order to make it almost equal to zero, we need to apply the limit. So,
Slope = limΔxà (f(x + Δx) – f(x)) / Δx
So, here the distance b/w the points is almost 0 and it will give an authentic value.
To Compute Definite Integrals:
Since definite integral is used to find out area under the curve. This area under the curve is b/w two fixed points. These two fixed points are limit points, where one point is known lower limit and second one is known as upper limit. It is important to observe that if curve is not smooth, we make partitions of the interval for which curve seems to be smooth for the accurate computation.
How to find limits?
In order to understand the concept of limits in a better way, some examples for the computation of limits are given below:
Example 1:
Compute limit of the function (7 − 2x + 10x2) at a point 2.
Solution:
Step 1: Rearrange the given function
Let f(x) = 10x2 – 2x + 7
Step 2: Apply limits i.e. x = 2
limxà2 f(x) = limxà2 (10x2 – 2x + 7)
Step 3: Simplify using appropriate rules.
limxà2 (10x2 – 2x + 7) = limxà2 (10x2) – limxà2 (2x) + limxà2 (7) [Sum & Difference rule]
limxà2 (10x2 – 2x + 7) = 10(2)2 – 2(2) + 7 [Constant & Constant Function rule]
limxà2 (10x2 – 2x + 7) = 40 – 4 + 7
limxà2 (10x2 – 2x + 7) = 43
Hence 43 is the required answer.
A limit solver can be used as an alternative to solve limits problems according to rules with steps.
Example 2
Compute limit of the function (8 + 2t) / (t2 +2) at point -3.
Solution:
Step 1: Rearrange the given function
Let f(x) = (2t + 8)/ (t2 + 2)
Step 2: Apply limits i.e. x = -3
limxà-3 f(x) = limxà-3 [(2t + 8)/ (t2 + 2)]
Step 3: Simplify using appropriate rules.
limxà-3 (2t + 8)/ (t2 + 2) = [limxà-3 (2t + 8)]/ [limxà-3 (t2 + 2)] [Quotient rule]
limxà-3 (2t + 8)/ (t2 + 2) = [limxà-3 2t + limxà-3 8] / [limxà-3 t2 + limxà-3 2] [Sum & Difference rule]
limxà-3 (8 + 2t)/ (t2 + 2) = [2(-3) + 8] / [(-3)2 + 2] [Constant & Constant Function rule]
limxà-3 (8 + 2t)/ (t2 + 2) = (– 6 + 8) / (9 + 2)
limxà-3 (8 + 2t)/ (t2 + 2) = 2/11 which is required answer.
Conclusion:
In this article, we elaborated on limits, their definition, and some important rules to compute limits. We also discussed useful applications of limits. In the last section, we solved examples to comprehend the concept of limit in a better way. We’re hopeful that after apprehending this article, you will be able to understand the concept of limit.
