Limits, Continuity, Differentiability and Discontinuities

Last Updated: 3rd May, 2025

In calculus, limits and continuity are important concepts that help us understand the behavior of functions as they approach certain values. A limit is a value that a function approaches as its input approaches a certain value. Continuity is the property of a function that describes whether it has any sudden jumps or breaks, or whether it can be drawn without lifting the pen from the paper.

Introduction to Limits and Continuity

• Limits and continuity are fundamental concepts in calculus. They allow us to understand how functions behave as their inputs approach certain values and to make precise statements about the behavior of functions at certain points.
• Continuity refers to the property of a function where the output changes smoothly and gradually as the input changes. A function is continuous at a point in the event that the limit of the function at that point exists and is equal to the esteem of the function at that point. In the event that a function isn't continuous at a point, it is said to be discontinuous.
• Limits, on the other hand, allude to the behavior of a function as the input approaches a certain value/point. A limit can be utilized to portray what happens to a function if the input gets subjectively near to a certain value, if the function isn't defined at that point.

Understanding Limits

The formal definition of a limit is given as follows:

For a function f(x), the limit of f(x) as x approaches a equals L, denoted as

lim f(x) = L  

for every number ε > 0 there exists a corresponding number δ > 0 such that |f(x) - L| < ε whenever 0 < |x - a| < δ.

This definition can be hard to understand at first, but it basically says that as x gets arbitrarily close to a, f(x) gets arbitrarily close to L. The ε and δ values represent arbitrarily small positive numbers that determine the level of precision required for the limit.

For example, consider the function

f(x) = (x^2 - 1)/(x - 1)

If we try to evaluate f(1), we get an undefined expression (0/0). However, we can find the limit of f(x) as x approaches 1 by factoring the numerator and simplifying:

= f(x) = (x^2 - 1)/(x - 1)
= (x + 1)(x - 1)/(x - 1)
= x + 1

Now we can see that as x approaches 1, f(x) approaches 2. Therefore, we can write lim f(x) as x approaches 1 = 2.

Properties of Limits

There are several important properties of limits that are useful for evaluating and manipulating functions. These include:

• The limit of a whole is the sum of the limits:

lim (f(x) + g(x)) = lim f(x) + lim g(x)

• The limit of a product is the product of the limits:

lim (f(x) * g(x)) = lim f(x) * lim g(x)

• The limit of a quotient is the quotient of the limits:

lim (f(x) / g(x)) = lim f(x) / lim g(x) (given lim g(x) ≠ 0)

• The limit of a constant multiple is the constant times the restrain:

lim (k * f(x)) = k * lim f(x)

• The limit of a composite function is the restrain of the external function connected to the constrain of the inner function:

lim f(g(x)) = f(lim g(x))

In expansion to these properties, there are a few hypotheses that can be utilized to assess limits, such as the squeeze theorem, which states that in the event that

g(x) ≤ f(x) ≤ h(x) 

for all x in a few interim containing a, and lim g(x) = lim h(x) = L, at that point lim f(x) = L. Another important theorem is the intermediate value theorem, which states that if f is a continuous function on the closed interval [a, b] and c is a number between f(a) and f(b), at that point there exists a number x between a and b such that f(x) = c.

Let's outline a few of these properties with a case. Consider the work

f(x) = x^2 - 3x + 2

We can use the properties of limits to evaluate lim f(x) as x approaches 2.

First, we can factor in the expression:

f(x) = x^2 - 3x + 2
         = (x - 1)(x - 2)

Using the product property of limits, we can write:

lim f(x) as x approaches 2 = lim (x - 1)(x - 2) as x approaches 2

Then, we can use the limit laws to simplify this expression:

= lim (x - 1)(x - 2) as x approaches 2
= (lim (x - 2)) * (lim (x - 1)) as x approaches 2
= (2 - 2) * (2 - 1)
=  0

Therefore, we can conclude that lim f(x) as x approaches 2 = 0.

Evaluating Limits

Several methods for evaluating limits include direct substitution, factoring, and L'Hopital's rule. Direct substitution is the simplest method, and it involves substituting the input value directly into the function and evaluating the resulting expression.

For example, to evaluate

lim (x^2 - 4x + 3)/(x - 3) 

as x approaches 3, we can simply substitute x = 3 into the expression to get:

lim (x^2 - 4x + 3)/(x - 3) = lim (3^2 - 4(3) + 3)/(3 - 3) = lim 0/0

Since this expression is indeterminate, we can use factoring or L'Hopital's rule to evaluate the limit further.

Factoring involves rewriting the function as a product of simpler expressions that can be canceled out.

For example, using the same limit as above, we can factor the numerator as

(x - 3)(x - 1) and cancel out the common factor of (x - 3) to get:

lim (x^2 - 4x + 3)/(x - 3) = lim (x - 1) 

as x approaches 3

This limit evaluates to 2, so we can conclude that

lim (x^2 - 4x + 3)/(x - 3) as x approaches 3 = 2

L'Hopital's rule is another method for evaluating indeterminate limits. It involves taking the derivative of both the numerator and denominator of the function and evaluating the resulting expression. For example, to evaluate lim sin(x)/x as x approaches 0, we can apply L'Hopital's rule to get:

lim sin(x)/x = lim cos(x) as x approaches 0

This limit evaluates to 1, so we can conclude that lim sin(x)/x as x approaches 0 = 1.

Continuity and Differentiability

• Continuity and differentiability are closely related concepts in calculus. A function is said to be continuous at a point if its graph has no breaks or jumps at that point. A function is said to be differentiable at a point if it has a well-defined tangent line at that point.
• The differentiability implies continuity theorem states that if a function is differentiable at a point, then it is also continuous at that point. However, the converse is not necessarily true - a function can be continuous at a point but not differentiable at that point.
• For example, the function f(x) = |x| is continuous at x = 0, but it is not differentiable at that point because it has a sharp corner.
• The relationship between continuity and differentiability can also be communicated regarding the derivative. In the event that a function is differentiable at a point, at that point its derivative exists at that point.
• The derivative gives the slant of the tangent line at that point, and the slant of the tangent line can be utilized to decide whether the function is expanding or diminishing at that point.

Types of Discontinuities

• A discontinuity occurs when a function has a break in its graph, either due to a hole, jump, or asymptote.
• A removable discontinuity happens when a function has a gap in its graph, which can be filled in by rethinking the function at that point.
• For illustration, the function f(x) = (x^2 - 4)/(x - 2) includes a removable discontinuity at x = 2, where the function is vague. Be that as it may, we are able to redefine the work as f(x) = x + 2 for x ≠ 2 to fill within the gap and make the function nonstop.
• A jump discontinuity happens when a work has two diverse limits from the left and right sides of a point. For illustration, the work f(x) = {x on the off chance that x < 0, 1 in case x ≥ 0} includes a jump discontinuity at x = 0, since the left-hand restrain is and the right-hand constrain is 1.
• An infinite discontinuity occurs when a function approaches positive or negative infinity at a certain point. For example, the function f(x) = 1/x has an infinite discontinuity at x = 0 since the function grows without bound as x approaches 0.

Applications of Limits and Continuity

Limits and continuity are important mathematical concepts that have a variety of applications in data science. Here are a few examples:

1. Derivatives and gradients: Limits are an essential component of calculus, which is the foundation of many data science techniques. Derivatives, which are based on limits, are used to calculate gradients, which are crucial for optimizing models using techniques such as gradient descent.
2. Probability distributions: Continuous probability distributions are often used in data science to model real-world phenomena. For example, the normal distribution is commonly used to model the distribution of heights or weights in a population.
3. Regression analysis: In regression analysis, a continuous variable is predicted based on other variables. The concept of continuity is essential here, as the prediction function must be continuous in order to be useful.
4. Time series analysis: Time series data is a sequence of values that are ordered by time. The continuity of time is an important aspect of time series analysis, as it allows us to model the relationship between past and future values.
5. Signal processing: Many signals, such as audio or video, are continuous in nature. Techniques from calculus, such as Fourier transforms, are used to analyze and process these signals.

Conclusion

In conclusion, limits and continuity are fundamental concepts in calculus that help us understand the behavior of functions as they approach certain values. They are important in many fields, such as physics, economics, and engineering, and are used to model and optimize various systems. The relationship between continuity and differentiability is also important, with differentiability implying continuity. Overall, understanding limits and continuity is crucial for a solid foundation in calculus.

Key Takeaways

• A limit is a value that a function approaches as its input approaches a certain value.
• Continuity is the property of a function that describes whether it has any sudden jumps or breaks.
• Evaluating limits can involve different methods, such as direct substitution, factoring, and L'Hopital's rule.
• Types of discontinuities include removable, jump, and infinite discontinuities.
• Differentiability implies continuity, but the converse is not necessarily true.
• Limits and continuity are fundamental concepts in calculus and are important for understanding the behaviour of functions.

Quiz

1. What is the definition of a limit in calculus?

1. The value that a function approaches as its input approaches a certain value 
2. The highest or lowest value of a function over a given interval
3. The derivative of a function at a certain point
4. The integral of a function over a certain interval

Answer: a. The value that a function approaches as its input approaches a certain value

2. What is a removable discontinuity?

1. A discontinuity where the limit of the function approaches infinity 
2. A discontinuity where the function has a jump or break, but can be fixed by changing the value of the function at that point 
3. A discontinuity where the function is undefined at a certain point 
4. A discontinuity where the function is not defined over a certain interval

Answer: b. A discontinuity where the function has a jump or break, but can be fixed by changing the value of the function at that point

3. If a function is differentiable at a certain point, what does that imply about its continuity?

1. The function is continuous at that point 
2. The function is discontinuous at that point 
3. The function is either continuous or discontinuous at that point, depending on the function 
4. The function may or may not be continuous at that point, depending on the function

Answer: a. The function is continuous at that point

4. What is L'Hopital's rule used for in calculus?

1. To find the area under a curve 
2. To evaluate limits of indeterminate forms, such as 0/0 or infinity/infinity 
3. To find the slope of a tangent line to a curve 
4. To determine whether a function is increasing or decreasing at a certain point

Answer: b. To evaluate limits of indeterminate forms, such as 0/0 or infinity/infinity