In calculus, limits are the most fundamental concept that is used to explain how a function behaves near a point. This is the basic idea of calculus that is used in various branches such as derivatives, integrals, and continuity.
In this post, we are going to explain the beginner’s guide to limits in calculus. We’ll provide the definition and notation of limit calculus first and then we’ll look at the rules, types, and calculations of limits.
What is the Limit in Calculus?
In mathematics, a calculus branch named limit is a value of a given function whose input gets closer and closer to the specific point, and the function never reaches that point in actuality. This branch of calculus is helpful for defining integral, differential, and continuity along with the usage to evaluate problems that involve infinite sequences and series.
Epsilon-Delta Definition of Limit Calculus
The epsilon-delta definition of this branch of calculus states that the limit of a function h(w) as the independent variable “w” approaches a specific point “p” equals to R if, for each positive number epsilon, there exists a positive number delta such that |f(x) – L| < epsilon whenever 0 < |x – c| < delta.
Notation of Limit Calculus
The notation of limits in calculus is a way of describing the behavior of a function at a given location. The most common notation is:
Limw→p h(w) = R
- This notation reads as “the limit of h(w) as w approaches p is equal to R”.
- The symbol “Lim” means “limit”
- The expression w→p means “w approaches p”,
- The expression h(w) is the function that we are taking the limit of.
- The letter R is the limit value.
Types of Limit Calculus
There are several types of limits in calculus. Here are some of them:
One-Sided Limits
One-sided limits are used to describe the behavior of a function as an independent variable “w” approaches a specific point “p” from only one side. There are two types of one-sided limits:
- left-hand limits
- right-hand limits
A left-hand limit of a function h(w) at a point “p” is the value that h(w) approaches as “w” approaches p from the left. In other words, it is the limit of h(w) as w approaches p from the values that are less than p.
The notation for a left-hand limit is:
Limw→p^- h(w) = R
A right-hand limit of a function h(w) at a point “p” is the value that h(w) approaches as “w” approaches p from the right. In other words, it is the limit of h(w) as w approaches p from the values that are greater than p.
The notation for a right-hand limit is:
Limw→p^+ h(w) = R
Two-Sided Limits
Two-sided limits are the most common type of limit. They are used to describe the behavior of a function as independent variable “w” approaches a specific point “p” from both the left and right sides.
Limw→p^- h(w) = Limw→p^+ h(w) = Limw→p h(w)
Infinite Limits
Infinite limits are limits that approach positive or negative infinity. They can be one-sided or two-sided. The expression for an infinite limit depends on the specific function and the direction of the limit.
Rules of Limit Calculus
Here are some of the most important rules of limit calculus:
Constant Rule
The limit value of a constant function remains the same at any specific point.
Limw→p K = K
Constant Multiple Rule
The limit of constant times a function is equal to the constant times the limit of the function.
Limw→p K h(w) = K Limw→p h(w)
Sum Rule
The limit of the sum of two functions is equal to the sum of the limits of the two functions.
Limw→p [h(w) + s(w)] = Limw→p h(w) + Limw→p s(w)
Difference Rule:
The limit of the difference between two functions is equal to the difference between the limits of the two functions.
Limw→p [h(w) – s(w)] = Limw→p h(w) – Limw→p s(w)
Product Rule:
The limit of the product of two functions is equal to the product of the limits of the two functions, as long as the limits of the two functions exist.
Limw→p [h(w) x s(w)] = Limw→p h(w) x Limw→p s(w)
Quotient Rule:
The limit of the quotient of two functions is equal to the quotient of the limits of the two functions, as long as the limits of the two functions exist and the denominator does not approach zero.
Limw→p [h(w) ÷ s(w)] = Limw→p h(w) ÷ Limw→p s(w)
Power Rule:
The limit of the nth power of a function is equal to the nth power of the limit of the function, as long as the limit of the function exists and n is a positive integer.
Limw→p [h(w)]n = [Limw→p [h(w)]n
How to Solve the Problems of Limit Calculus?
There are many ways to find the limit. Some of the most common methods include:
Direct Substitution:
If the function is defined at the point where the limit is being evaluated, then the limit is equal to the value of the function at that point.
Example
Evaluate the limit of h(w) = 3w2 + 4w3 + 12w if “w” approaches “3”.
Solution
Step 1: Write the function h(w) as the mathematical representation of the limit.
Limw→p [h(w)] = Limw→3 [3w2 + 4w3 + 12w]
Step 2: Now simplify the function using the sum and difference rules.
Limw→3 [3w2 + 4w3 + 12w] = Limw→3 [3w2] + Limw→3 [4w3] + Limw→3 [12w]
Step 3: Now take out the constant terms outside the notation of limit.
Limw→3 [3w2 + 4w3 + 12w] = 3Limw→3 [w2] + 4Limw→3 [w3] + 12Limw→3 [w]
Step 4: Now use the direct substitution method and substitute w = 3 in the above expression.
Limw→3 [3w2 + 4w3 + 12w] = 3 [32] + 4 [33] + 12 [3]
Limw→3 [3w2 + 4w3 + 12w] = 3 [9] + 4 [27] + 12 [3]
Limw→3 [3w2 + 4w3 + 12w] = 27 + 108 + 36
Limw→3 [3w2 + 4w3 + 12w] = 171
Factoring:
If the function is a rational function, then the limit can be evaluated by factoring the numerator and denominator.
Example
Evaluate the limit as “w” approaches 6 of the function h(w) = (w2 – 36) / (w – 6)
Solution
Step 1: Write the function h(w) as the mathematical representation of the limit.
Limw→p [h(w)] = Limw→6 [(w2 – 36) / (w – 6)]
Step 2: Now simplify the function using the sum and difference rules.
Limw→6 [(w2 – 36) / (w – 6)] = (Limw→6 [w2] – Limw→6 [36]) / (Limw→6 [w] – Limw→6 [6])
Limw→6 [(w2 – 36) / (w – 6)] = ([62] – [36]) / ([6] – [6])
Limw→6 [(w2 – 36) / (w – 6)] = (36 – 36) / (6 – 6)
Limw→6 [(w2 – 36) / (w – 6)] = 0/0
The function is in an indeterminate form. We can use the factoring method to remove the indeterminate form.
Step 3: Now make the factors of the numerator and substitute it to the original function.
w2 – 36 = w2 – (6)2
w2 – 36 = (w – 6) (w + 6)
Limw→6 [(w2 – 36) / (w – 6)] = Limw→6 [(w – 6) (w + 6) / (w – 6)]
Step 4: Now simplify the above expression and apply the limit again.
Limw→6 [(w2 – 36) / (w – 6)] = Limw→6 [(w – 6) (w + 6) / (w – 6)]
Limw→6 [(w2 – 36) / (w – 6)] = Limw→6 (w + 6)
Limw→6 [(w2 – 36) / (w – 6)] = Limw→6 (w) + Limw→6 (6)
Limw→6 [(w2 – 36) / (w – 6)] = 6 + 6
Limw→6 [(w2 – 36) / (w – 6)] = 12
Rationalize the Numerator:
If the function is a rational function with a denominator that includes a square root, then the limit can be evaluated by rationalizing the numerator.
Example
Evaluate the limit as “w” approaches 49 of the function h(w) = (√w – 7) / (w – 49)
Solution
Step 1: Write the function h(w) as the mathematical representation of the limit.
Limw→p [h(w)] = Limw→49 [(√w – 7) / (w – 49)]
Step 2: Now use the direct substitution method and substitute t = 49 in the above expression.
Limw→49 [(√w – 7) / (w – 49)] = (Limw→49 [√w] – Limw→49 [7]) / (Limw→49 [w] – Limw→49 [49])
Limw→49 [(√w – 7) / (w – 49)] = ([√49] – [7]) / ([49] – [49])
Limw→49 [(√w – 7) / (w – 49)] = (7 – 7) / (49 – 49)
Limw→49 [(√w – 7) / (w – 49)] = 0/0
The function is in an indeterminate form. We can use rationalizing the numerator to remove the indeterminate form.
Step 3: Now rationalize the given function
(√w – 7) / (w – 49) = [(√w – 7) / (w – 49)] x [(√w + 7) / (√w + 7)]
(√w – 7) / (w – 49) = [(√w – 7) (√w + 7)] / [(w – 49) (√w + 7)]
(√w – 7) / (w – 49) = [(√w)2 – (7)2] / [(w – 49) (√w + 7)]
(√w – 7) / (w – 49) = [w – 49] / [(w – 49) (√w + 7)]
Step 4: Now simplify the above expression.
(√w – 7) / (w – 49) = [w – 49] / [(w – 49) (√w + 7)]
(√w – 7) / (w – 49) = 1/(√w + 7)
Step 5: Now apply the limits again
Limt→49 [(√w – 7) / (w – 49)] = Limt→49 [1/(√w + 7)]
Limt→49 [(√w – 7) / (w – 49)] = Limt→49 [1] / (Limt→49 [√w] + Limt→49 [7])
Limt→49 [(√w – 7) / (w – 49)] = [1] / ([√49] + [7])
Limt→49 [(√w – 7) / (w – 49)] = 1 / (7 + 7)
Limt→49 [(√w – 7) / (w – 49)] = 1 / 14
Additional Tips for Solving Limit Problems
Here are some additional tips for solving problems of limit calculus:
- Use the rules of limits whenever possible.
- Try to simplify the function as much as possible before evaluating the limit.
- If you are stuck, try to find a similar problem that you have solved before, or use a limit calculator to help you solve the problem.
Conclusion
In this article, we have introduced the basics of limits in calculus. We have defined limits, discussed the epsilon-delta definition of a limit, and looked at the properties of limits. We have also seen some examples of limits.
If you are interested in learning more about limits, there are many great resources available online and in libraries. With a little practice, you will be able to understand and use limits to solve calculus problems.
