Chapter 2: Sections 2.3 & 2.5 - Delving into Limits
Welcome back to Professor Baker's Math Class! Today, we're tackling Sections 2.3 and 2.5, which focus on understanding and calculating limits. Let's break down the key concepts and work through some examples to solidify your understanding.
2.3: Calculating Limits Using the Limit Laws
The limit laws provide a structured approach to finding limits of complex functions by breaking them down into simpler components. Remember, these laws apply only if the individual limits exist. Here's a summary of the essential limit laws:
- Sum Law: The limit of a sum is the sum of the limits: $$lim_{x \to a} [f(x) + g(x)] = lim_{x \to a} f(x) + lim_{x \to a} g(x)$$
- Difference Law: The limit of a difference is the difference of the limits: $$lim_{x \to a} [f(x) - g(x)] = lim_{x \to a} f(x) - lim_{x \to a} g(x)$$
- Constant Multiple Law: The limit of a constant times a function is the constant times the limit of the function: $$lim_{x \to a} [cf(x)] = c lim_{x \to a} f(x)$$
- Product Law: The limit of a product is the product of the limits: $$lim_{x \to a} [f(x)g(x)] = lim_{x \to a} f(x) \cdot lim_{x \to a} g(x)$$
- Quotient Law: The limit of a quotient is the quotient of the limits (provided the limit of the denominator is not zero): $$lim_{x \to a} \frac{f(x)}{g(x)} = \frac{lim_{x \to a} f(x)}{lim_{x \to a} g(x)}, \text{ if } lim_{x \to a} g(x) \neq 0$$
- Power Law: The limit of a function raised to a power is the limit of the function raised to that power: $$lim_{x \to a} [f(x)]^n = [lim_{x \to a} f(x)]^n$$, where n is a positive integer.
- Root Law: The limit of a root of a function is the root of the limit of the function: $$lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{lim_{x \to a} f(x)}$$, where n is a positive integer (if n is even, we assume that $$lim_{x \to a} f(x) > 0$$).
- Constant Limit: The limit of a constant is the constant itself: $$lim_{x \to a} c = c$$
- Identity Limit: $$lim_{x \to a} x = a$$
Let's see an example of applying these laws:
Example: Evaluate $$lim_{x \to 5} (2x^2 - 3x + 4)$$.
Using the limit laws, we can break this down:
$$lim_{x \to 5} (2x^2 - 3x + 4) = lim_{x \to 5} 2x^2 - lim_{x \to 5} 3x + lim_{x \to 5} 4$$
$$ = 2 lim_{x \to 5} x^2 - 3 lim_{x \to 5} x + 4$$
$$ = 2(5^2) - 3(5) + 4 = 50 - 15 + 4 = 39$$
2.5: Continuity
A function $f$ is continuous at a number $a$ if the following three conditions are met:
- $f(a)$ is defined (i.e., $a$ is in the domain of $f$).
- $\lim_{x \to a} f(x)$ exists.
- $\lim_{x \to a} f(x) = f(a)$.
In simpler terms, a function is continuous at a point if there are no breaks, jumps, or holes at that point. You can draw the graph of the function through that point without lifting your pen!
Keep practicing, and you'll become a limit law master! Good luck with your studies!