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Get Started Now- Lesson: 110:24
- Lesson: 23:55
- Lesson: 3a7:25
- Lesson: 49:43
- Lesson: 510:17

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 $\lim_{x \to 0} \;\frac{{{sin\;}5x}}{{2x}}$

24.

Limits

24.1

Finding limits from graphs

24.2

Limit laws

24.3

Continuity

24.4

Finding limits algebraically - direct substitution

24.5

Finding limits algebraically - when direct substitution is not possible

24.6

Infinite limits - vertical asymptotes

24.7

Limits at infinity - horizontal asymptotes

24.8

Intermediate value theorem

24.9

Squeeze theorem

We have over 860 practice questions in Sixth Year Maths for you to master.

Get Started Now24.1

Finding limits from graphs

24.3

Continuity

24.4

Finding limits algebraically - direct substitution

24.5

Finding limits algebraically - when direct substitution is not possible

24.6

Infinite limits - vertical asymptotes

24.7

Limits at infinity - horizontal asymptotes

24.8

Intermediate value theorem

24.9

Squeeze theorem