One Sided Limits and Continuity

One-Sided Limits and Continuity

In calculus, one-sided limits and continuity are essential concepts that help in understanding the behavior of functions near specific points. These concepts are used to describe how functions behave as they approach a point from one direction (either from the left or from the right), and how functions behave in general in terms of smoothness and connectedness.

One-Sided Limits

A one-sided limit refers to the limit of a function as the input approaches a certain point, but only from one side—either from the left or from the right.

• Right-hand limit: The limit as $x$ approaches $c$ from values greater than $c$ (from the right side).
• Left-hand limit: The limit as $x$ approaches $c$ from values smaller than $c$ (from the left side).

Notation:

• Right-hand limit:

• Left-hand limit:

Interpretation of One-Sided Limits:

• The right-hand limit describes how the function behaves as $x$ gets closer to $c$ from values greater than $c$ (i.e., from the right).
• The left-hand limit describes how the function behaves as $x$ gets closer to $c$ from values less than $c$ (i.e., from the left).

The limit exists at $c$ only if the left-hand and right-hand limits are equal.

Example of One-Sided Limits

Consider the function:

Let's find the left-hand and right-hand limits as $x \to 2$.

1. Right-hand limit (approaching 2 from the right, i.e., $x \to 2^+$):

Since $x \geq 2$ for values approaching 2 from the right, the function behaves as $f(x) = x + 1$. Therefore:

2. Left-hand limit (approaching 2 from the left, i.e., $x \to 2^-$):

Since $x < 2$ for values approaching 2 from the left, the function behaves as $f(x) = 2x - 1$. Therefore:

3. Conclusion: Since the left-hand and right-hand limits are both equal to 3, we can say:

Continuity

A function is said to be continuous at a point $c$ if the following three conditions are satisfied:

1. The function is defined at $c$: The function has a value at the point $c$.

   $$f(c) \text{ is defined.}$$

2. The limit exists at $c$: The limit of the function as $x$ approaches $c$ from both the left and right exists.

   $$\lim_{x \to c} f(x) \text{ exists.}$$

3. The limit equals the function value: The value of the function at $c$ is equal to the limit as $x$ approaches $c$.

   $$\lim_{x \to c} f(x) = f(c)$$

If these three conditions are satisfied, the function is continuous at $c$.

Types of Discontinuities

A function may fail to be continuous at a point. In such cases, we have different types of discontinuities:

1. Jump Discontinuity: The left-hand and right-hand limits at $c$ exist but are not equal. The function "jumps" at the point.

   Example: A step function or piecewise function where the value changes abruptly.

2. Infinite Discontinuity: The function approaches infinity (or negative infinity) as $x$ approaches $c$. This often happens when the denominator of a rational function becomes 0.

   Example: $f(x) = \frac{1}{x}$ at $x = 0$.

3. Removable Discontinuity: The left-hand and right-hand limits exist and are equal, but the function is either not defined at $c$ or the value of the function at $c$ does not match the limit.

   Example: $f(x) = \frac{x^2 - 4}{x - 2}$, where the limit at $x = 2$ is 4, but the function is not defined at $x = 2$.

Example of Continuity

Let's consider the function:

We need to determine whether this function is continuous at $x = 2$.

1. The function is defined at $x = 2$: From the function definition, we see that $f(2) = 4$.

2. The limit exists at $x = 2$: We calculate the limit of $f(x)$ as $x$ approaches 2. Since the function is $x^2$ for $x \neq 2$, we find:

3. The limit equals the function value: Since $f(2) = 4$ and $\lim_{x \to 2} f(x) = 4$, we conclude that the function is continuous at $x = 2$.

Thus, $f(x)$ is continuous at $x = 2$.

Discontinuity Example

Consider the function:

1. The function is defined at $x = 2$: From the function definition, $f(2) = 5$.

2. The limit exists at $x = 2$: To find the limit, we simplify the expression $\frac{x^2 - 4}{x - 2}$:

Thus,

3. The limit does not equal the function value: Since $\lim_{x \to 2} f(x) = 4$ but $f(2) = 5$, the function is not continuous at $x = 2$.

This is an example of a removable discontinuity, as the limit exists but does not match the function value.

Summary

• One-sided limits describe the behavior of a function as it approaches a point from one side (either left or right).
• A function is continuous at a point if:
  1. It is defined at the point.
  2. The limit exists at the point.
  3. The limit equals the function value at that point.
• Discontinuities can be classified as:
  • Jump discontinuity: Left-hand and right-hand limits exist but are not equal.
  • Infinite discontinuity: The function tends to infinity at a point.
  • Removable discontinuity: The limit exists but is not equal to the function value at that point.