The Limit as the Instantaneous Rate of Change or Slope of the Tangent Line

The limit represents the instantaneous rate of change of a function, equivalent to the slope of the tangent line at a point.

Key concept: As x approaches a specific value, f(x) approaches a specific value L.

Applying Factoring to Evaluate Limits of Rational Functions

When direct substitution yields the indeterminate form 0/0:

1. Factor numerator and denominator
2. Cancel common factors
3. Apply direct substitution

Example: $\lim_{x \to 2}\frac{x^2-4}{x-2} = \lim_{x \to 2}(x+2) = 4$

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 f(x) arbitrarily close to L by taking x sufficiently close to c (but not equal to c).

• Examine behavior as x approaches from both left and right
• Look for holes, jumps, and asymptotes
• Check if left-hand limit equals right-hand limit

Use values of f(x) for x-values approaching c from both sides:

• If f(x) approaches the same value from both sides -> limit exists
• If values differ or approach ±∞ -> limit does not exist (DNE)

The two-sided limit $\lim_{x \to c}f(x)$ exists if and only if: $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)$

Examining how function behaves as x approaches a specific value:

• From the left: $\lim_{x \to c^-} f(x)$
• From the right: $\lim_{x \to c^+} f(x)$
• Two-sided limit exists only if both one-sided limits are equal

1. Jump discontinuity: Left and right limits exist but are different
2. Infinite discontinuity: Function approaches ±∞
3. Oscillating discontinuity: Function oscillates wildly (e.g., $\sin(1/x)$ near 0)

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

• Sum: $\lim_{x \to c}[f(x) + g(x)] = L + M$
• Difference: $\lim_{x \to c}[f(x) - g(x)] = L - M$
• Product: $\lim_{x \to c}[f(x) \cdot g(x)] = L \cdot M$
• Quotient: $\lim_{x \to c}\frac{f(x)}{g(x)} = \frac{L}{M}$ (if M ≠ 0)
• Constant multiple: $\lim_{x \to c}[k \cdot f(x)] = k \cdot L$
• Power: $\lim_{x \to c}[f(x)]^n = L^n$

If f is continuous at x = c, then: $\lim_{x \to c} f(x) = f(c)$

Works for: polynomials, rational functions (where defined), trig, exponential, and log functions at continuous points.

Factoring: For 0/0 forms involving polynomials

• Example: $\lim_{x \to 1}\frac{x^2-1}{x-1} = \lim_{x \to 1}(x+1) = 2$

Conjugates: For 0/0 forms involving radicals

• Multiply numerator and denominator by conjugate
• Example: $\lim_{x \to 0}\frac{\sqrt{x+4}-2}{x} \cdot \frac{\sqrt{x+4}+2}{\sqrt{x+4}+2} = \lim_{x \to 0}\frac{x}{x(\sqrt{x+4}+2)} = \frac{1}{4}$

If $g(x) \leq f(x) \leq h(x)$ near c and $\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$, then $\lim_{x \to c} f(x) = L$.

Special limits: $\lim_{x \to 0}\frac{\sin x}{x} = 1, \quad \lim_{x \to 0}\frac{1-\cos x}{x} = 0$

• Removable (hole): Limit exists but ≠ f(c) or f(c) undefined
• Jump: Left-hand limit ≠ right-hand limit
• Infinite: One or both one-sided limits are infinite

A function f is continuous at x = c if ALL THREE conditions are met:

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

1. Does f(c) exist? (function defined at c)
2. Does the limit exist? (left-hand = right-hand limit)
3. Does the limit equal the function value?

f is continuous on [a, b] if:

• Continuous on (a, b)
• Continuous from the right at a
• Continuous from the left at b

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

Vertical asymptote at x = c if one or both one-sided limits are infinite.

$\lim_{x \to \infty} f(x) = L \quad \text{or} \quad \lim_{x \to -\infty} f(x) = L$

Horizontal asymptote at y = L if limit exists and is finite.

Analyzing Horizontal Asymptotes Using Limits at Infinity

For rational functions, compare degrees:

• Numerator degree < denominator degree: $\lim_{x \to \pm\infty} = 0$
• Degrees equal: $\lim_{x \to \pm\infty} = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$
• Numerator degree > denominator degree: $\lim_{x \to \pm\infty} = \pm\infty$

If f is continuous on [a, b] and k is between f(a) and f(b), then there exists at least one c ∈ (a, b) such that f(c) = k.

Requirements:

1. f must be continuous on [a, b]
2. f(a) ≠ f(b)

Corollary: If f is continuous on [a, b] and f(a) - f(b) < 0 (opposite signs), then there exists c ∈ (a, b) such that f(c) = 0.

Used to prove existence of solutions to equations.