Limits: Definition, Techniques, Important Results, & Examples

Limits are the key concept of mathematics that provide valuable insights into how functions behave at singular points and around them. Limits are used in physics, engineering, economics, and various other fields to model and analyze various phenomena involving change and growth.

Limit is an important concept in calculus and is used in almost all significant and major concepts, including differentiation and integrals. A limit is a point or degree beyond which something cannot expand. A limit represents something that restricts, constrains, or binds.

In this article, we’ll demystify the concept of limits, explore techniques to solve limit problems, delve into important results, and provide illustrative examples that will enhance your understanding.

What is the Limit?

The limit of a function is the approximate value of a function. The limit of a function specifies the behavior of a function as it approaches to a specific point.The mathematical notation used for expressing limits involves the use of the limit symbol and variables that tend towards a certain value. For instance, the limit of a function f(x) as x approaches a is denoted as:

lim_x→a f(x)

Graphically, the limit of a function can be visualized by observing the behavior of the function’s graph as it gets closer and closer to a particular point. This helps in understanding how the function behaves near that point.

Techniques for Solving Limit Problems:

Here we elaborate on some useful techniques that help to solve the problems of limits.

1.   Direct Substitution:

When the function is well-defined at the given point, direct substitution involves directly substituting the value into the function to find the limit.

2.   Factoring and Cancelling:

Factoring and canceling techniques are useful for simplifying complex expressions, making them more amenable to limit calculations.

3.   Rationalizing Techniques:

Rationalizing techniques are particularly useful when dealing with expressions involving radicals or fractions.

4.   Squeeze Theorem:

The Squeeze Theorem provides a way to evaluate limits by “sandwiching” a function between two other functions with known limits.

5.   L’Hôpital’s Rule:

L’Hôpital’s Rule is very useful specifically comes in handy for resolving limits involving indeterminate forms like 0/0 or ∞/∞ encountered in limit problems. It states that the limit of the ratio of two functions is equivalent to the limit of their derivatives.

6.   Limit calculator

Using a limit calculator is an easier way to find the limit of a function at a given point. All you need to do is input the function and the specific point, and the solution with steps will be provided in just a few seconds.

Important Results:

Here we will elaborate on some important deductions to remember that are very useful and employed during the calculations of limit problems.

• Lim _{n à + ∞} (1 + 1/n)ⁿ = e
• Lim _{x à 0} [(e^x – 1) / x] = log_e e = 1
• Lim _{x à 0} [(a^x – 1) / x] = log_e a
• Lim _{x à 0} (1 + x)^1/x = e
• Lim _{x à – ∞} e^x = Lim_{xà – ∞} (1/e^-x) = 0
• Lim _{xà ± ∞} (a/x) = 0, where a is any real number.
• Lim _{θà 0} sin θ/ θ = 1 if θ is measured in radian.

Examples of Finding Limits

Finding limits is a fundamental concept in calculus. Here are some basic examples of finding limits, along with their solutions:

Example 1:

Express the limit in terms of the number ‘e’ Lim_{xà + ∞} (1 + 4/n)²ⁿ.

Solution:

Step 1:Given data

Lim_{xà + ∞} (1 + 4/n)²ⁿ

Step 2:Compare the given problem with the following result.

Lim _{n à + ∞} (1 + 1/n)ⁿ = e

Step 3:Now

(1 + 4/n)²ⁿ = [(1 + 4/n)^n/4]⁸ = [(1 + 1/n/4)^n/4]⁸

Lim_{xà + ∞} (1 + 4/n)²ⁿ = Lim_{xà + ∞} [(1 + 1/m)^m]⁸  (put m = n/4, when n à∞ then, n à∞)

Lim_xà+∞ (1 + 4/n)²ⁿ = e⁸

Example 2:

Evaluate Lim_θà0 sin (5θ)/ θ

Solution:

Step 1:Given data

Lim _θà0 sin(5θ)/ θ

Step 2:Compare the given problem with the following fundamental result.

Lim _θà0 sin (θ)/ θ = 1

Step 3:Now

Let x = 5θ so that θ = x/5

When θ à 0, we have x à 0

So,

Lim _{θà 0}sin 5θ/ θ = Lim _{xà 0} (sin x)/(x/5) = 5Lim _{xà 0} sin x/ x

Lim _{θà 0} sin 5θ/ θ = 5 (1) = 5

Example 3:

Compute the limit of the function f(x) = (x² – 16) / (x – 4) as x approaches 4 applying suitable limit rules.

Solution:

Step 1:Given data

Function = f(x) = [(x² – 16) / (x – 4)] as x approaches 4.

Step 2: Simplify, applying the limit value.

lim_xà4 f(x) = lim_xà4 [(x² – 16) / (x – 4)]

lim_xà4 f(x) = lim_xà4 (x² – 16) / lim_xà4 (x – 4)     (Quotient rule)

lim_xà4 f(x) = [(4)^{^2} – 16) / (4 – 4)]

lim_xà4 f(x) = (16 – 16) / (4 – 4)

lim_xà4 f(x) = 0 / 0

As we have encountered that this function answers an indeterminate 0/0 form. So, first of all, we will simplify this.

Step 3:Simplification

lim_xà4 f(x) = lim_xà4 [(x² – 16) / (x – 4)]

lim_xà4 f(x) = lim_xà4 [(x² – 4²) / (x – 4)]

lim_xà4 f(x) = lim_xà4 [(x + 4) * (x – 4) / (x – 4)]

lim_xà4 f(x) = lim_xà4 (x + 4)

Step 4:Place the values

lim_xà4 f(x) = lim_xà4 x + lim_xà4 4   (sum rule)

lim_xà4 f(x) = 4 + 4

lim_xà4f(x) = 8

Summary:

Limits are a basis of mathematics especially it is the core dimension of calculus. In this blog, we have explored the concept of limits in mathematics. We have elaborated on the definition of limit, some useful techniques, and important results.We have also solved some examples to apprehend the concept of the limit in a more precise and concise way.