Limits, Continuity and Differentiability

Limits

In mathematics, a limit describes the value that a function or sequence "approaches" as the input or index approaches some value. Limits are fundamental to calculus and mathematical analysis. Understanding limits is the first step towards understanding continuity and differentiability.

Definition of a Limit

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

Intuitive Understanding of Limits

Imagine you are walking towards a specific point on a graph. The limit of the function at that point is the height (y-value) you are approaching as you get closer and closer to that point. It doesn't matter what the actual value of the function is *at* that point, or even if the function is defined there. What matters is the trend as you approach.

Types of Limits

There are several types of limits to consider:

• Right-hand Limit: The limit as x approaches 'c' from values greater than 'c'. Denoted as $$ \lim_{x \to c^+} f(x) $$
• Left-hand Limit: The limit as x approaches 'c' from values less than 'c'. Denoted as $$ \lim_{x \to c^-} f(x) $$
• Two-sided Limit: For the two-sided limit to exist, both the left-hand and right-hand limits must exist and be equal. $$ \lim_{x \to c} f(x) = L \quad \text{if and only if} \quad \lim_{x \to c^-} f(x) = L \quad \text{and} \quad \lim_{x \to c^+} f(x) = L $$
• Limits at Infinity: This describes the behavior of the function as x increases or decreases without bound. Denoted as $$ \lim_{x \to \infty} f(x) $$ or $$ \lim_{x \to -\infty} f(x) $$

Properties of Limits

If $$ \lim_{x \to c} f(x) = L $$ and $$ \lim_{x \to c} g(x) = M $$, then:

• Sum Rule: $$ \lim_{x \to c} [f(x) + g(x)] = L + M $$
• Difference Rule: $$ \lim_{x \to c} [f(x) - g(x)] = L - M $$
• Constant Multiple Rule: $$ \lim_{x \to c} [k \cdot f(x)] = k \cdot L $$ (where k is a constant)
• Product Rule: $$ \lim_{x \to c} [f(x) \cdot g(x)] = L \cdot M $$
• Quotient Rule: $$ \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L}{M} $$ (provided $$ M \neq 0 $$)
• Power Rule: $$ \lim_{x \to c} [f(x)]^n = L^n $$ (where n is a positive integer)
• Root Rule: $$ \lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{L} $$ (provided L is positive when n is even)

Evaluating Limits

There are several methods to evaluate limits:

1. Direct Substitution: If the function is continuous at 'c', you can simply substitute 'c' into the function.
2. Factoring and Cancelling: If direct substitution results in an indeterminate form (like $$ \frac{0}{0} $$), try factoring the numerator and denominator to cancel out common factors.
3. Multiplying by the Conjugate: Useful for limits involving square roots. Multiply the numerator and denominator by the conjugate of the expression.
4. L'Hôpital's Rule: If direct substitution results in an indeterminate form $$ \frac{0}{0} $$ or $$ \frac{\infty}{\infty} $$, you can take the derivative of the numerator and the derivative of the denominator separately and then evaluate the limit. $$ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} $$ (provided the limit on the right exists).
5. Squeeze Theorem (Sandwich Theorem): If $$ g(x) \le f(x) \le h(x) $$ for all x near 'c' (except possibly at 'c'), and $$ \lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L $$, then $$ \lim_{x \to c} f(x) = L $$.

Example of Factoring and Cancelling

Evaluate $$ \lim_{x \to 2} \frac{x^2 - 4}{x - 2} $$.

Direct substitution gives $$ \frac{2^2 - 4}{2 - 2} = \frac{0}{0} $$, which is indeterminate.

Factor the numerator: $$ x^2 - 4 = (x - 2)(x + 2) $$.

So, $$ \lim_{x \to 2} \frac{(x - 2)(x + 2)}{x - 2} $$.

Cancel the $$ (x - 2) $$ terms (since $$ x \to 2 $$, $$ x \neq 2 $$, so $$ x - 2 \neq 0 $$):

$$ \lim_{x \to 2} (x + 2) $$.

Now, substitute $$ x = 2 $$: $$ 2 + 2 = 4 $$.

Therefore, $$ \lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4 $$.

Continuity

A function is continuous at a point if its graph can be drawn without lifting the pen. Formally, a function $$ f(x) $$ is continuous at a point $$ x = c $$ if three conditions are met:

1. $$ f(c) $$ is defined. (The function exists at the point).
2. $$ \lim_{x \to c} f(x) $$ exists. (The limit exists as x approaches the point).
3. $$ \lim_{x \to c} f(x) = f(c) $$ (The limit equals the function value at the point).

Types of Discontinuities

If any of the above conditions are not met, the function is discontinuous at $$ x = c $$. There are several types of discontinuities:

• Removable Discontinuity: Occurs when $$ \lim_{x \to c} f(x) $$ exists, but either $$ f(c) $$ is undefined or $$ f(c) \neq \lim_{x \to c} f(x) $$. This is often seen in cases where a factor can be cancelled. The discontinuity can be "removed" by redefining the function at that point.
• Jump Discontinuity: Occurs when the left-hand limit and the right-hand limit exist but are not equal ($$ \lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x) $$). This is common in piecewise functions.
• Infinite Discontinuity: Occurs when at least one of the one-sided limits is $$ \infty $$ or $$ -\infty $$. This typically happens when the denominator of a rational function approaches zero, leading to a vertical asymptote.
• Oscillating Discontinuity: Occurs when a function oscillates infinitely between two values as x approaches 'c', so the limit does not exist.

Continuity on an Interval

A function is continuous on an open interval $$ (a, b) $$ if it is continuous at every point in the interval. For a closed interval $$ [a, b] $$, the function must be continuous on $$ (a, b) $$ and the following one-sided limits must exist and equal the function values:

• $$ \lim_{x \to a^+} f(x) = f(a) $$
• $$ \lim_{x \to b^-} f(x) = f(b) $$

Properties of Continuous Functions

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

• $$ f + g $$
• $$ f - g $$
• $$ k \cdot f $$ (where k is a constant)
• $$ f \cdot g $$
• $$ \frac{f}{g} $$ (provided $$ g(c) \neq 0 $$)

Furthermore, if $$ g $$ is continuous at $$ c $$ and $$ f $$ is continuous at $$ g(c) $$, then the composite function $$ f(g(x)) $$ is continuous at $$ c $$.

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 $$ f(c) = N $$.

This theorem is useful for proving the existence of roots for equations. For example, if $$ f(a) $$ is negative and $$ f(b) $$ is positive, the IVT guarantees that the function must cross the x-axis (i.e., have a root) somewhere between $$ a $$ and $$ b $$.

Example of Continuity

Consider the function: $$ f(x) = \begin{cases} x^2 + 1 & \text{if } x < 1 \\ 3 & \text{if } x = 1 \\ 2x + 1 & \text{if } x > 1 \end{cases} $$ Check for continuity at $$ x = 1 $$.

1. Is $$ f(1) $$ defined? Yes, $$ f(1) = 3 $$.
2. Does $$ \lim_{x \to 1} f(x) $$ exist? We need to check the left-hand and right-hand limits.
   • Left-hand limit: $$ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (x^2 + 1) = 1^2 + 1 = 2 $$.
   • Right-hand limit: $$ \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (2x + 1) = 2(1) + 1 = 3 $$.
   Since the left-hand limit (2) is not equal to the right-hand limit (3), the limit $$ \lim_{x \to 1} f(x) $$ does not exist.
3. Does $$ \lim_{x \to 1} f(x) = f(1) $$? Since the limit does not exist, this condition is not met.

Conclusion: The function $$ f(x) $$ is discontinuous at $$ x = 1 $$. It has a jump discontinuity because the one-sided limits exist but are unequal.

Differentiability

A function is differentiable at a point if its derivative exists at that point. The derivative of a function at a point represents the instantaneous rate of change of the function, or the slope of the tangent line to the function's graph at that point.

Definition of the Derivative

The derivative of a function $$ f(x) $$ at a point $$ x = c $$, denoted by $$ f'(c) $$, is defined as: $$ f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h} $$ This is also known as the limit definition of the derivative. An alternative form is: $$ f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c} $$

Geometric Interpretation

The derivative $$ f'(c) $$ is the slope of the line tangent to the curve $$ y = f(x) $$ at the point $$ (c, f(c)) $$. For the derivative to exist, the tangent line must be unique and non-vertical.

Relationship between Continuity and Differentiability

A crucial theorem states: If a function $$ f $$ is differentiable at $$ x = c $$, then it must be continuous at $$ x = c $$.

However, the converse is NOT always true: A function can be continuous at a point but NOT differentiable at that point.

Conditions for Non-Differentiability

A function $$ f $$ is NOT differentiable at $$ x = c $$ if:

• $$ f $$ is discontinuous at $$ x = c $$.
• The graph of $$ f $$ has a "sharp corner" or "cusp" at $$ x = c $$. This occurs when the limit of the difference quotient from the left is different from the limit from the right. ($$ \lim_{h \to 0^-} \frac{f(c+h) - f(c)}{h} \neq \lim_{h \to 0^+} \frac{f(c+h) - f(c)}{h} $$).
• The graph of $$ f $$ has a vertical tangent line at $$ x = c $$. This occurs when the limit of the difference quotient approaches $$ \infty $$ or $$ -\infty $$.

Differentiability on an Interval

A function is differentiable on an open interval $$ (a, b) $$ if it is differentiable at every point in the interval.

Rules of Differentiation (Brief Overview)

While the limit definition is fundamental, we use differentiation rules for efficiency:

• Power Rule: $$ \frac{d}{dx}(x^n) = nx^{n-1} $$
• Constant Rule: $$ \frac{d}{dx}(k) = 0 $$
• Constant Multiple Rule: $$ \frac{d}{dx}[k \cdot f(x)] = k \cdot f'(x) $$
• Sum/Difference Rule: $$ \frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x) $$
• Product Rule: $$ \frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x) $$
• Quotient Rule: $$ \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} $$
• Chain Rule: $$ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) $$

Example of Differentiability

Consider the function $$ f(x) = |x| $$. Is it differentiable at $$ x = 0 $?

First, check continuity at $$ x = 0 $$.

1. $$ f(0) = |0| = 0 $$. Defined.
2. $$ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (-x) = 0 $$.
3. $$ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (x) = 0 $$.
4. Since $$ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = 0 $$, $$ \lim_{x \to 0} f(x) = 0 $$.
5. $$ \lim_{x \to 0} f(x) = f(0) $$. The function is continuous at $$ x = 0 $$.

Now, check differentiability using the limit definition: $$ f'(0) = \lim_{h \to 0} \frac{f(0+h) - f(0)}{h} = \lim_{h \to 0} \frac{|h| - 0}{h} = \lim_{h \to 0} \frac{|h|}{h} $$

We need to check the left-hand and right-hand limits:

• Left-hand limit: $$ \lim_{h \to 0^-} \frac{|h|}{h} = \lim_{h \to 0^-} \frac{-h}{h} = \lim_{h \to 0^-} (-1) = -1 $$.
• Right-hand limit: $$ \lim_{h \to 0^+} \frac{|h|}{h} = \lim_{h \to 0^+} \frac{h}{h} = \lim_{h \to 0^+} (1) = 1 $$.

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

Conclusion: The function $$ f(x) = |x| $$ is continuous at $$ x = 0 $$ but not differentiable at $$ x = 0 $$. This is because the graph has a sharp corner (a "V" shape) at the origin.

Summary Table: Limits, Continuity, Differentiability