Introduction to Limits
In this lecture, we will cover the basics of limits, focusing on how to evaluate them both analytically and graphically.
Evaluating Limits Analytically
- Direct Substitution: Attempt to substitute the value directly into the function.
- Example: Limit as x approaches 2 of
(x^2 - 4)/(x - 2) results in 0/0, which is undefined.
- If direct substitution results in an indeterminate form, try values close to the desired point.
- Calculate f(2.1), f(2.01), etc., to find that the limit approaches 4.
- Factoring: Simplify expressions to remove indeterminacy.
- Example: Factor
x^2 - 4 to (x+2)(x-2). Cancel (x-2) and find limit as x approaches 2 of x+2, which equals 4.
- Non-Fraction Case: Use direct substitution if there is no fraction or indeterminacy.
- Example: Limit as x approaches 5 of
x^2 + 2x - 4 equals 31 using direct substitution.
- Difference of Cubes: Use formulas like
a^3 - b^3 = (a-b)(a^2 + ab + b^2) to factor expressions.
- Example: Factor
x^3 - 27 and evaluate the limit.
- Complex Fractions: Simplify by multiplying numerator and denominator by common denominators and/or conjugates.
- Example: Simplify
1/x - 1/3 over x-3 by multiplying by 3x, then use substitution.
- Radicals: Multiply by conjugates to remove radicals.
- Example: Simplify
(sqrt(x) - 3)/(x-9) by multiplying by the conjugate sqrt(x) + 3.
Evaluating Limits Graphically
- One-Sided Limits:
- Determine the limit as x approaches a point from the left or right by following the curve on the graph.
- Example: As
x approaches -3 from the left, the limit is 1; from the right, it is -3.
- If the one-sided limits are not equal, the overall limit does not exist.
- Function Value: Check the closed circle on the graph for the function value.
- Example: At
x=-3, the function value is where the closed circle is, say y=-3.
Types of Discontinuities
- Hole (Removable Discontinuity): Occurs when a factor can be canceled out.
- Jump Discontinuity: Non-removable, seen when the graph jumps from one value to another.
- Infinite Discontinuity: Occurs at vertical asymptotes, also non-removable.
- Example:
1/(x-3) results in a vertical asymptote at x=3.
These strategies help in evaluating limits analytically and graphically, providing a foundation for further study in calculus.