Section 2.4

Computing Limits Algebraically

The algebraic toolkit for resolving indeterminate forms and finding exact limit values.

1

The Algebra of Limits

The Limit Laws allow us to distribute limits across arithmetic operations, provided the component limits exist.

Sum Rule
Product Rule
Quotient Rule
Power Rule

Direct Substitution Property: For polynomials and rational functions (with non-zero denominator), .

2

Resolving : Factoring

When direct substitution results in , it indicates that the numerator and denominator share a common factor that becomes zero at the limit point.

Example

Factor the numerator:
Cancel the common factor (valid since we're NOT at ):
3

The Conjugate Method

For limits involving radicals, we rationalize the expression to "free" the variable from the root.

Example:

Step 1: Multiply by the conjugate
Step 2: Simplify numerator
Step 3: Cancel and evaluate
4

Complex Fractions

Example:

Combine the numerator into a single fraction:
Cancel and evaluate:
5

The Squeeze Theorem

Theorem Statement

If near , and

Then as well.

The Archetypal Example:

Step 1: Bound the oscillating part
Step 2: Multiply by (positive)
Step 3: Apply Squeeze Theorem

Since and ,

Deep Insight

The Squeeze Theorem proves by squeezing a sector's area between two triangles. Without this, we cannot derive the derivatives of sine and cosine!

6

Interactive Visualization

Squeeze Theorem Visualization

Watch how f(x) = x²sin(1/x) oscillates wildly but is trapped between x² and -x², forcing the limit to 0.

f(x) = x²sin(1/x) - the squeezed function
h(x) = x² - upper bound
g(x) = -x² - lower bound
8

Practice Quiz

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