Limits and Continuity
Grasp the concept of infinity and approaching values.
The Concept of a Limit
In calculus, a limit is the value that a function "approaches" as the input "approaches" some value. Limits are essential to calculus and are used to define derivatives, integrals, and continuity.
What happens to the function f(x) as x gets closer and closer to a certain number?
An Intuitive Introduction: Upper Bounds and The Method of Exhaustion
Zeno's Dichotomy Paradox
Imagine you need to walk to a wall. First, you cover half the distance. Then, you cover half of the *remaining* distance. Then half of *that* remainder, and so on. Zeno's paradox is that you will never technically reach the wall, as you always have a small distance left to halve.
This introduces the idea of a limit and an **upper bound**. While you take infinitely many steps, the total distance you've traveled gets closer and closer to the width of the room. The distance to the wall is the **limit** of the sum of the distances you've walked. The total distance is always less than the width of the room, so the room's width is an **upper bound**.
The Area of a Circle
The ancient Greeks used a similar idea, the "method of exhaustion," to determine the area of a circle. They understood how to find the area of regular polygons. By inscribing a polygon inside a circle, they could see its area was *less than* the circle's area.
As they increased the number of sides of the polygon (from a triangle to a square, to a pentagon, and so on), the polygon's area would "exhaust" the area of the circle, getting closer and closer to the true value. The area of the circle is the limit of the area of the inscribed polygon as the number of sides approaches infinity.
Condition for the Existence of a Limit
For a limit to exist at a particular point, the function must approach the same value from both the left side and the right side. These are known as one-sided limits.
- Left-Hand Limit: The value the function approaches as x gets closer to c from values *less than* c. This is denoted as
limx→c⁻ f(x). - Right-Hand Limit: The value the function approaches as x gets closer to c from values *greater than* c. This is denoted as
limx→c⁺ f(x).
The limit of f(x) as x approaches c exists if and only if the left-hand limit equals the right-hand limit.
limx→c⁻ f(x) = limx→c⁺ f(x)If the left and right-hand limits are not equal, we say the limit does not exist at that point.
Indeterminate Quantities
When evaluating limits, you will often encounter forms like 0/0 or ∞/∞. These are called indeterminate quantities because you cannot determine their value from the form alone. A limit with an indeterminate form could be any number, or it might be ∞, -∞, or not exist at all.
Proof: 0/0 is indeterminate
0 = 0 * any no. => 0/0 = any no.. This proves that 0/0 is an indeterminate quantity
We can also show this with a few examples:
- Case 1: Limit is 2. Let f(x) = 2x and g(x) = x. Then lim (x→0) f(x) = 0 and lim (x→0) g(x) = 0. The limit of their quotient is lim (x→0) 2x/x = lim (x→0) 2 = 2.
- Case 2: Limit is 0. Let f(x) = x² and g(x) = x. The limit of their quotient is lim (x→0) x²/x = lim (x→0) x = 0.
Since the form 0/0 can lead to different results, it is indeterminate. Any quantity that can be expressed in the form 0/0 is also indeterminate
Proof: ∞/∞ is indeterminate
The form ∞/∞ can be reduced to 0/0. k/∞ = 0 => ∞ = k/0, where k is any finite no. So ∞/∞ = (k1/0)/(k2/0) => ∞/∞ = (k1/0)*(0/k2) => ∞/∞ = 0/0 Hence it is proved that ∞/∞ is also indeterminate.
Similarly for the form ∞/∞:
- Case 1: Limit is 2. Let f(x) = 2x and g(x) = x. As x→∞, both functions approach ∞. Then lim (x→∞) 2x/x = lim (x→∞) 2 = 2.
- Case 2: Limit is ∞. Let f(x) = x² and g(x) = x. Then lim (x→∞) x²/x = lim (x→∞) x = ∞.
- Case 3: Limit is 0. Let f(x) = x and g(x) = x². Then lim (x→∞) x/x² = lim (x→∞) 1/x = 0.
Again, the form ∞/∞ can result in different values, proving it is indeterminate.
Proof: ∞ - ∞ is indeterminate
k/∞ = 0 => ∞ = k/0, where is any finite no. So ∞ - ∞ = (k1/0) - (k2/0) => ∞ - ∞ = ((k1*0) - (k2*0))/0 => ∞ - ∞ = 0/0 It is hence proved that ∞ - ∞ is also indeterminate
Consider the limit of f(x) - g(x) as x→∞.
- Case 1: Limit is 0. Let f(x) = x and g(x) = x. Then lim (x→∞) (x - x) = 0.
- Case 2: Limit is 5. Let f(x) = x + 5 and g(x) = x. Then lim (x→∞) (x + 5 - x) = 5.
- Case 3: Limit is ∞. Let f(x) = 2x and g(x) = x. Then lim (x→∞) (2x - x) = lim (x→∞) x = ∞.
The form ∞ - ∞ is indeterminate.
Proof: 1∞ is indeterminate
Let x = 1∞. Taking log on both sides, log x = ∞ log 1 => log x = (k/0)*0 => log x = 0/0 => x = antilog(0/0). Therefore x is indeterminate.
This is a famous indeterminate form. Consider lim (x→∞) (1 + 1/x)x. This limit is the very definition of the number e ≈ 2.718. However, consider lim (x→∞) (1 + 2/x)x. This limit evaluates to e². Since the form can lead to different results, it is indeterminate.
Proof: ∞0 is indeterminate
Let x = ∞0. Taking log on both sides, log x = 0 log ∞ => log x = 0 * (-k/0) as log 0 -> -∞ => log x = 0/0. It is hence proved that x is indeterminate.
Consider lim (x→∞) f(x)g(x).
- Case 1: Limit is 1. Let f(x) = x and g(x) = 1/x. We evaluate lim (x→∞) x1/x. Taking the logarithm, we get lim (x→∞) ln(x)/x, which by L'Hopital's rule is 0. So the original limit is e⁰ = 1.
- Case 2: Limit is e. Let f(x) = ex and g(x) = 1/x. Then lim (x→∞) (ex)1/x = lim (x→∞) e = e.
The form ∞0 is indeterminate.
Proof: 0∞ is NOT indeterminate
Let x = 0∞. Taking log on both sides, log x = ∞ * log 0 => log x = ∞ * (-∞) as log 0 -> -∞ => log x = -∞ => x = 0.
Let's analyze the form lim (x→c) f(x)g(x), where lim f(x) = 0 and lim g(x) = ∞. This means we are multiplying a number that is getting closer and closer to zero by itself an infinite number of times.
For example, (0.1)10 is very small. (0.1)100 is even smaller. As the exponent approaches infinity, the result will approach 0 faster and faster. No matter how the base approaches 0 or the exponent approaches ∞, the result will always be 0.
Therefore, 0∞ = 0 and is a determinate form.
Therefore, 0∞ = 0 and is a determinate form.
Examples of Calculating Limits
Continuity
A function is continuous if you can draw its graph without lifting your pen from the paper. This means there are no holes, jumps, or vertical asymptotes.
Formally, a function f(x) is continuous at a point x = c if:
- f(c) is defined.
- The limit of f(x) as x approaches c exists.
- The limit of f(x) as x approaches c equals f(c).
Combining these conditions, for a function to be continuous at a point `c`, we must have:
limx→c⁻ f(x) = limx→c⁺ f(x) = f(c)Types of Discontinuity
When a function is not continuous at a point, we say it has a discontinuity. There are three main types:
Removable Discontinuity
This occurs when the limit of the function exists at a point, but the function is either not defined at that point, or the function's value is different from the limit. It looks like a "hole" in the graph.
limx→c⁻ f(x) = limx→c⁺ f(x) ≠ f(c)Infinite Discontinuity
This occurs when one or both of the one-sided limits go to infinity or negative infinity. This is seen as a vertical asymptote in the graph.
limx→c⁻ f(x) = ±∞ OR limx→c⁺ f(x) = ±∞Jump Discontinuity
This happens when the left-hand and right-hand limits both exist but are not equal. The graph "jumps" from one value to another.
limx→c⁻ f(x) ≠ limx→c⁺ f(x)