MathematicsChapter 2: Limit, Continuity and Derivative
Infinite Limits and Vertical Asymptotes
If magnitude grows without bound near x=a, x=a is a vertical asymptote; signs may differ by side.
Learning objectives
- State the rule for infinite limits and vertical asymptotes.
- Solve a infinite limits and vertical asymptotes problem step by step.
- Check the result, restrictions, notation, and units.
Definition
If magnitude grows without bound near x=a, x=a is a vertical asymptote; signs may differ by side.
Formula
x=a is a vertical asymptote
Where / symbols
- Infinite limits and vertical asymptotes
- If magnitude grows without bound near x=a, x=a is a vertical asymptote; signs may differ by side.
CONCEPT AND RULE
If magnitude grows without bound near x=a, x=a is a vertical asymptote; signs may differ by side.
METHOD
For infinite limits and vertical asymptotes, identify the given information and required result. Select the matching definition or formula, state its conditions, substitute carefully, simplify one line at a time, and check.
WORKED EXAMPLE
1x-2 tends to -infinity from the left and +infinity from the right.
COMMON MISTAKE
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
How to solve infinite limits and vertical asymptotes
- Identify the mathematical objects, givens, restrictions, and required result.
- Choose and write the relevant definition, theorem, or formula.
- Substitute with brackets and carry out one justified step per line.
- Simplify, state the final answer clearly, and verify it in the original setting.
Worked example
- Worked example:
- 1x-2 tends to -infinity from the left and +infinity from the right.
In simple words
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Key points
- Infinite limits and vertical asymptotes
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Infinite Limits and Vertical Asymptotes
Sticky Notes · Formula and Worked Revision
Core Idea
- If magnitude grows without bound near x=a, x=a is a vertical asymptote; signs may differ by side.
Rule / Formula
- If magnitude grows without bound near x=a, x=a is a vertical asymptote; signs may differ by side.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- 1x-2 tends to -infinity from the left and +infinity from the right.
Common Error
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
FBISE Exam Tip
- Show the formula, substitutions, algebra, restrictions, and final answer; method marks need visible reasoning.
60-second revision
- If magnitude grows without bound near x=a, x=a is a vertical asymptote; signs may differ by side.
- Worked pattern: 1x-2 tends to -infinity from the left and +infinity from the right.
- Final check: domain, sign, notation, units, and reasonableness.
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