Limit Laws
Limits of sums, products and quotients are computed from the parts.
Key idea
Limits of sums, products and quotients are computed from the parts.
In depth
Limits of sums, differences, products, quotients (denominator limit not zero) and constant multiples can be computed from the limits of the pieces. These laws let you evaluate limits of complicated expressions by direct substitution when functions are well behaved.
Worked example
lim (3x² + 2x) as x → 2 = 3(4) + 2(2) = 16.
Formula
Real-world uses
- Computing the long-run cost of a process built from simple parts in engineering and finance
Common mistake
Applying the quotient law when the denominator tends to zero.
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 Limit Laws?
Limits of sums, products and quotients are computed from the parts. It is a topic at the advanced level in Calculus.
What is the formula for Limit Laws?
The key formula is lim (f+g) = lim f + lim g.
Where is Limit Laws used in real life?
Real-world examples: Computing the long-run cost of a process built from simple parts in engineering and finance.
What should I learn before Limit Laws?
Make sure you are comfortable with One-Sided Limits, the previous topic in Calculus.
What is a common mistake with Limit Laws?
Applying the quotient law when the denominator tends to zero.
How can I practise Limit Laws?
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 Limit Laws?
Move on to Factoring and Rationalizing Limits 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.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- 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.