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Get Started Now- Lesson: 110:24
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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 $\lim_{x \to 0} \;\frac{{{sin\;}5x}}{{2x}}$

16.

Limits

16.1

Finding limits from graphs

16.2

Continuity

16.3

Finding limits algebraically - direct substitution

16.4

Finding limits algebraically - when direct substitution is not possible

16.5

Infinite limits - vertical asymptotes

16.6

Limits at infinity - horizontal asymptotes

16.7

Intermediate value theorem

16.8

Squeeze theorem

We have over 700 practice questions in Higher 2 Maths for you to master.

Get Started Now16.1

Finding limits from graphs

16.2

Continuity

16.3

Finding limits algebraically - direct substitution

16.4

Finding limits algebraically - when direct substitution is not possible

16.5

Infinite limits - vertical asymptotes

16.6

Limits at infinity - horizontal asymptotes

16.7

Intermediate value theorem

16.8

Squeeze theorem