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There are times when applying direct substitution would only give us an undefined solution. In this section, we will explore some cool tricks to evaluate limits algebraically, such as using conjugates, trigonometry, common denominators, and factoring.

- 1.
**Simplify Out "Zero Denominator" by Cancelling Common Factors**Find $\lim_{x \to 3} \;\frac{{{x^2} - 9}}{{x - 3}}$

- 2.
**Expand First, Then Simplify Out "Zero Denominator" by Cancelling Common Factors**Evaluate $\lim_{h \to 0} \;\frac{{{{\left( {5 + h} \right)}^2} - 25}}{h}$

- 3.
**Simplify Out "Zero Denominator" by Rationalizing Radicals**Evaluate:

a)$\lim_{x \to 4} \;\frac{{4 - x}}{{2 - \sqrt x }}$*(hint: rationalize the denominator by multiplying its conjugate)* - 4.
**Find Limits of Functions involving Absolute Value**Evaluate $\lim_{x \to 0} \;\frac{{\left| x \right|}}{x}$

*(hint: express the absolute value function as a piece-wise function)* - 5.
**Find Limits Using the Trigonometric Identity:$\lim_{\theta \to 0} \;\frac{{{sin\;}\theta}}{{\theta}}=1$**Find $\lim_{x \to 0} \;\frac{{{sin\;}5x}}{{2x}}$

1.

Limits

1.1

Introduction to Calculus - Limits

1.2

Finding limits from graphs

1.3

Limit laws

1.4

Continuity

1.5

Finding limits algebraically - direct substitution

1.6

Finding limits algebraically - when direct substitution is not possible

1.7

Infinite limits - vertical asymptotes

1.8

Limits at infinity - horizontal asymptotes

1.9

Intermediate value theorem

1.10

Squeeze theorem

1.2

Finding limits from graphs

1.4

Continuity

1.5

Finding limits algebraically - direct substitution

1.6

Finding limits algebraically - when direct substitution is not possible

1.7

Infinite limits - vertical asymptotes

1.8

Limits at infinity - horizontal asymptotes

1.9

Intermediate value theorem

1.10

Squeeze theorem