Limits and Continuity

Introduction to Limits

In mathematics, a limit describes the value that a function "approaches" as the input to the function approaches some value. Limits are fundamental to calculus and mathematical analysis. They are used to define concepts such as continuity, derivatives, and integrals.

Consider a function $f(x)$. We are interested in the behavior of $f(x)$ as $x$ gets closer and closer to a specific value, say $c$. The limit of $f(x)$ as $x$ approaches $c$, denoted as $\lim_{x \to c} f(x)$, is the value that $f(x)$ tends to reach. It's important to note that the limit does not depend on the value of the function at $c$ itself, or even if the function is defined at $c$.

Definition of a Limit (Informal)

We say that the limit of $f(x)$ as $x$ approaches $c$ is $L$, written as $\lim_{x \to c} f(x) = L$, if we can make the values of $f(x)$ arbitrarily close to $L$ by choosing $x$ sufficiently close to $c$, but not equal to $c$.

Examples of Limits

Example 1: Linear Function

Let $f(x) = 2x + 3$. What is the limit as $x$ approaches 4?

As $x$ gets closer to 4, $2x$ gets closer to $2 \times 4 = 8$. So, $2x + 3$ gets closer to $8 + 3 = 11$. Therefore, $\lim_{x \to 4} (2x + 3) = 11$.

Example 2: Quadratic Function

Let $f(x) = x^2$. What is the limit as $x$ approaches 3?

As $x$ gets closer to 3, $x^2$ gets closer to $3^2 = 9$. Therefore, $\lim_{x \to 3} x^2 = 9$.

Example 3: Function with a Hole

Let $f(x) = \frac{x^2 - 4}{x - 2}$. What is the limit as $x$ approaches 2?

If we try to substitute $x=2$, we get $\frac{2^2 - 4}{2 - 2} = \frac{0}{0}$, which is an indeterminate form. However, we can simplify the function for $x \neq 2$: $f(x) = \frac{(x-2)(x+2)}{x-2} = x+2$. Now, as $x$ approaches 2 (but is not equal to 2), $f(x)$ approaches $2+2=4$. So, $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4$.

Left-Hand and Right-Hand Limits

Sometimes, the behavior of a function as it approaches a point from the left (values less than the point) might be different from its behavior as it approaches from the right (values greater than the point).

• Left-Hand Limit: The limit of $f(x)$ as $x$ approaches $c$ from values less than $c$. It is denoted as $\lim_{x \to c^-} f(x)$.
• Right-Hand Limit: The limit of $f(x)$ as $x$ approaches $c$ from values greater than $c$. It is denoted as $\lim_{x \to c^+} f(x)$.

For the overall limit $\lim_{x \to c} f(x)$ to exist, the left-hand limit and the right-hand limit must exist and be equal. That is, $\lim_{x \to c} f(x) = L$ if and only if $\lim_{x \to c^-} f(x) = L$ and $\lim_{x \to c^+} f(x) = L$.

Example: Piecewise Function

Consider the function: $f(x) = \begin{cases} x + 1 & \text{if } x < 0 \\ x^2 & \text{if } x \ge 0 \end{cases}$

Let's find the limits as $x$ approaches 0.

Left-Hand Limit: As $x$ approaches 0 from the left ($x < 0$), $f(x) = x + 1$. $\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (x + 1) = 0 + 1 = 1$.

Right-Hand Limit: As $x$ approaches 0 from the right ($x \ge 0$), $f(x) = x^2$. $\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x^2 = 0^2 = 0$.

Since the left-hand limit (1) is not equal to the right-hand limit (0), the overall limit $\lim_{x \to 0} f(x)$ does not exist.

Indeterminate Forms

When evaluating a limit by direct substitution, we sometimes encounter indeterminate forms. These forms do not give us enough information to determine the limit, and we need to use other techniques like algebraic manipulation, L'Hôpital's Rule, or series expansion. Common indeterminate forms include:

• $\frac{0}{0}$
• $\frac{\infty}{\infty}$
• $0 \times \infty$
• $\infty - \infty$
• $1^\infty$
• $0^0$
• $\infty^0$

Techniques for Evaluating Limits

1. Direct Substitution

If the function $f(x)$ is continuous at $x=c$, then $\lim_{x \to c} f(x) = f(c)$. This is the simplest method and applies to polynomials, rational functions (where the denominator is non-zero), trigonometric functions, exponential functions, and logarithmic functions at points where they are defined.

2. Algebraic Manipulation (Factoring, Rationalizing, Common Denominator)

This is used when direct substitution leads to an indeterminate form like $\frac{0}{0}$.

Example: Factoring

Evaluate $\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$. Direct substitution gives $\frac{1^2 - 1}{1 - 1} = \frac{0}{0}$. Factor the numerator: $\frac{(x-1)(x+1)}{x-1}$. Cancel out $(x-1)$ (since $x \to 1$, $x \neq 1$): $x+1$. Now substitute: $\lim_{x \to 1} (x+1) = 1+1 = 2$.

Example: Rationalizing

Evaluate $\lim_{x \to 0} \frac{\sqrt{x+1} - 1}{x}$. Direct substitution gives $\frac{\sqrt{0+1} - 1}{0} = \frac{1-1}{0} = \frac{0}{0}$. Multiply numerator and denominator by the conjugate of the numerator: $\frac{\sqrt{x+1} - 1}{x} \times \frac{\sqrt{x+1} + 1}{\sqrt{x+1} + 1} = \frac{(x+1) - 1}{x(\sqrt{x+1} + 1)} = \frac{x}{x(\sqrt{x+1} + 1)}$. Cancel out $x$: $\frac{1}{\sqrt{x+1} + 1}$. Now substitute: $\lim_{x \to 0} \frac{1}{\sqrt{x+1} + 1} = \frac{1}{\sqrt{0+1} + 1} = \frac{1}{1+1} = \frac{1}{2}$.

Example: Common Denominator

Evaluate $\lim_{x \to 0} \left( \frac{1}{x} - \frac{1}{x^2} \right)$. Direct substitution gives $\infty - \infty$. Find a common denominator: $\lim_{x \to 0} \frac{x - 1}{x^2}$. Now substitute: $\frac{0-1}{0^2} = \frac{-1}{0}$. This indicates the limit approaches $\pm \infty$. Consider the sign: As $x \to 0$, $x^2$ is always positive. The numerator $x-1$ approaches -1. So, $\lim_{x \to 0} \frac{x - 1}{x^2} = -\infty$.

3. Using Standard Limits

Certain limits are frequently used and should be memorized.

• $\lim_{x \to 0} \frac{\sin x}{x} = 1$
• $\lim_{x \to 0} \frac{\tan x}{x} = 1$
• $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$
• $\lim_{x \to 0} \frac{a^x - 1}{x} = \ln a$ (for $a > 0$)
• $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$
• $\lim_{x \to 0} (1+x)^{1/x} = e$
• $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$
• $\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$

4. L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool for evaluating limits that result in indeterminate forms $\frac{0}{0}$ or $\frac{\infty}{\infty}$. If $\lim_{x \to c} \frac{f(x)}{g(x)}$ yields $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then: $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$ provided the limit on the right exists or is $\pm \infty$. We can apply this rule repeatedly as long as the indeterminate form persists.

Example: Using L'Hôpital's Rule

Evaluate $\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}$. Direct substitution gives $\frac{e^0 - 1 - 0}{0^2} = \frac{1 - 1 - 0}{0} = \frac{0}{0}$. Apply L'Hôpital's Rule (differentiate numerator and denominator): $\lim_{x \to 0} \frac{\frac{d}{dx}(e^x - 1 - x)}{\frac{d}{dx}(x^2)} = \lim_{x \to 0} \frac{e^x - 1}{2x}$. Direct substitution again gives $\frac{e^0 - 1}{2(0)} = \frac{1-1}{0} = \frac{0}{0}$. Apply L'Hôpital's Rule again: $\lim_{x \to 0} \frac{\frac{d}{dx}(e^x - 1)}{\frac{d}{dx}(2x)} = \lim_{x \to 0} \frac{e^x}{2}$. Now substitute: $\frac{e^0}{2} = \frac{1}{2}$. So, $\lim_{x \to 0} \frac{e^x - 1 - x}{x^2} = \frac{1}{2}$.

Introduction to Continuity

Intuitively, a function is continuous if its graph can be drawn without lifting the pen from the paper. Mathematically, continuity at a point means that the limit of the function exists at that point, the function is defined at that point, and the limit equals the function's value.

Definition of Continuity at a Point

A function $f(x)$ is said to be continuous at a point $x=c$ if all three of the following conditions are met:

1. $f(c)$ is defined.
2. $\lim_{x \to c} f(x)$ exists.
3. $\lim_{x \to c} f(x) = f(c)$.

Types of Discontinuities

If any of the conditions for continuity at a point $x=c$ are not met, the function is said to be discontinuous at $c$. There are several types of discontinuities:

1. Removable Discontinuity

This occurs when $\lim_{x \to c} f(x)$ exists, but either $f(c)$ is undefined or $\lim_{x \to c} f(x) \neq f(c)$. This type of discontinuity can be "removed" by redefining the function at $x=c$ to be equal to the limit.

Example:

The function $f(x) = \frac{x^2 - 4}{x - 2}$ has a removable discontinuity at $x=2$. We found $\lim_{x \to 2} f(x) = 4$. If we define $f(2) = 4$, the function becomes continuous at $x=2$.

2. Jump Discontinuity

This occurs when the left-hand limit and the right-hand limit at $x=c$ both exist, but are not equal ($\lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x)$).

Example:

The piecewise function $f(x) = \begin{cases} x + 1 & \text{if } x < 0 \\ x^2 & \text{if } x \ge 0 \end{cases}$ has a jump discontinuity at $x=0$ because $\lim_{x \to 0^-} f(x) = 1$ and $\lim_{x \to 0^+} f(x) = 0$.

3. Infinite Discontinuity

This occurs when at least one of the one-sided limits is infinite (i.e., $\lim_{x \to c^-} f(x) = \pm \infty$ or $\lim_{x \to c^+} f(x) = \pm \infty$). This typically happens when the denominator of a rational function approaches zero. The line $x=c$ is a vertical asymptote.

Example:

The function $f(x) = \frac{1}{x}$ has an infinite discontinuity at $x=0$. $\lim_{x \to 0^-} \frac{1}{x} = -\infty$ $\lim_{x \to 0^+} \frac{1}{x} = +\infty$ The line $x=0$ (the y-axis) is a vertical asymptote.

4. Oscillating Discontinuity

This occurs when a function oscillates infinitely many times as $x$ approaches $c$, so that the limit does not exist.

Example:

Consider the function $f(x) = \sin(\frac{1}{x})$ as $x \to 0$. As $x$ gets closer to 0, $\frac{1}{x}$ becomes very large (positive or negative). The sine function oscillates between -1 and 1 infinitely often in any interval around 0. Therefore, $\lim_{x \to 0} \sin(\frac{1}{x})$ does not exist, and the function has an oscillating discontinuity at $x=0$.

Continuity on an Interval

A function $f(x)$ is said to be continuous on an open interval $(a, b)$ if it is continuous at every point $c$ in the interval.

A function $f(x)$ is said to be continuous on a closed interval $[a, b]$ if it is continuous on the open interval $(a, b)$, and additionally, the right-hand limit at $a$ equals $f(a)$ ($\lim_{x \to a^+} f(x) = f(a)$), and the left-hand limit at $b$ equals $f(b)$ ($\lim_{x \to b^-} f(x) = f(b)$).

Properties of Continuous Functions

If $f$ and $g$ are continuous functions at $x=c$, then the following functions are also continuous at $x=c$:

• Sum: $(f+g)(x) = f(x) + g(x)$
• Difference: $(f-g)(x) = f(x) - g(x)$
• Product: $(f \cdot g)(x) = f(x) \cdot g(x)$
• Quotient: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, provided $g(c) \neq 0$.

Also, if $f$ is continuous at $c$ and $g$ is continuous at $f(c)$, then the composite function $(g \circ f)(x) = g(f(x))$ is continuous at $c$.

Important Theorems Related to Continuity

1. Intermediate Value Theorem (IVT)

If $f$ is continuous on the closed interval $[a, b]$, and $N$ is any number between $f(a)$ and $f(b)$ (where $f(a) \neq f(b)$), then there exists at least one number $c$ in the open interval $(a, b)$ such that $f(c) = N$.

2. Extreme Value Theorem (EVT)

If $f$ is continuous on a closed interval $[a, b]$, then $f$ attains both an absolute maximum value and an absolute minimum value on $[a, b]$. These extreme values occur either at the endpoints ($a$ or $b$) or at critical points within the interval $(a, b)$.

Limits at Infinity

We often need to understand the behavior of a function as the input $x$ becomes very large (approaches positive infinity, $x \to \infty$) or very small (approaches negative infinity, $x \to -\infty$).

The limit $\lim_{x \to \infty} f(x) = L$ means that the values of $f(x)$ can be made arbitrarily close to $L$ by choosing $x$ sufficiently large and positive.

The limit $\lim_{x \to -\infty} f(x) = M$ means that the values of $f(x)$ can be made arbitrarily close to $M$ by choosing $x$ sufficiently large and negative.

These limits are particularly important for determining horizontal asymptotes of a function's graph.

Evaluating Limits at Infinity for Rational Functions

For a rational function $f(x) = \frac{P(x)}{Q(x)}$, where $P(x) = a_n x^n + \dots + a_0$ and $Q(x) = b_m x^m + \dots + b_0$:

• If the degree of the numerator ($n$) is less than the degree of the denominator ($m$) ($n < m$), then $\lim_{x \to \pm \infty} f(x) = 0$.
• If the degree of the numerator equals the degree of the denominator ($n = m$), then $\lim_{x \to \pm \infty} f(x) = \frac{a_n}{b_m}$ (the ratio of the leading coefficients).
• If the degree of the numerator is greater than the degree of the denominator ($n > m$), then the limit is either $\infty$, $-\infty$, or does not exist.

Example: Limits at Infinity

Evaluate $\lim_{x \to \infty} \frac{3x^2 + 2x - 1}{5x^2 - x + 4}$. The degree of the numerator (2) equals the degree of the denominator (2). The limit is the ratio of the leading coefficients: $\frac{3}{5}$.

Evaluate $\lim_{x \to -\infty} \frac{x^3 - 2x + 5}{2x^4 + 1}$. The degree of the numerator (3) is less than the degree of the denominator (4). The limit is 0.

Evaluate $\lim_{x \to \infty} \frac{4x^3 + x - 1}{2x^2 + 3x}$. The degree of the numerator (3) is greater than the degree of the denominator (2). To analyze further, divide by the highest power of $x$ in the denominator ($x^2$): $\lim_{x \to \infty} \frac{4x + \frac{1}{x} - \frac{1}{x^2}}{2 + \frac{3}{x}}$. As $x \to \infty$, the terms with $x$ in the denominator go to 0. The expression behaves like $\frac{4x}{2} = 2x$. So, $\lim_{x \to \infty} \frac{4x^3 + x - 1}{2x^2 + 3x} = \infty$.

Continuity and Differentiability Connection

A crucial relationship exists between continuity and differentiability. If a function is differentiable at a point $c$, then it must also be continuous at $c$.

However, the converse is not true: a function can be continuous at a point but not differentiable there. A common example is the absolute value function $f(x) = |x|$ at $x=0$. It is continuous at $x=0$, but it has a sharp corner, making it non-differentiable at $x=0$.