Factoring and Rationalizing Limits
Remove 0/0 indeterminacy with algebra.
Key idea
Remove 0/0 indeterminacy with algebra.
In depth
When substitution gives 0/0, the expression hides a common factor. Factor and cancel it, or multiply by a conjugate to remove a root, then substitute.
Worked example
lim (x² − 9)/(x − 3) = lim (x + 3) = 6. And lim (√(x + 1) − 1)/x = lim 1/(√(x + 1) + 1) = 1/2.
Real-world uses
- Finding the value a formula settles to when substituting directly gives 0/0
Common mistake
Cancelling terms rather than factors, or concluding 0/0 means no limit.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Factoring and Rationalizing Limits?
Remove 0/0 indeterminacy with algebra. It is a topic at the advanced level in Calculus.
Where is Factoring and Rationalizing Limits used in real life?
Real-world examples: Finding the value a formula settles to when substituting directly gives 0/0.
What should I learn before Factoring and Rationalizing Limits?
Make sure you are comfortable with Limit Laws, the previous topic in Calculus.
What is a common mistake with Factoring and Rationalizing Limits?
Cancelling terms rather than factors, or concluding 0/0 means no limit.
How can I practise Factoring and Rationalizing Limits?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Factoring and Rationalizing Limits?
Move on to Limits at Infinity next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Limits at Infinity – End behaviour of functions.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.
- Squeeze Theorem – If g ≤ f ≤ h and g, h → L then f → L.