Te reo Māori terms
click each term to open in Te AkaLearning objective overview
Master Teaching Guide coverage for this objective.
Resources
- Other: Not in Beta Textbook; no verified external resource found (this NCEA Level 3 calculus content -- evaluating limits by direct substitution, factor-cancel on 0/0 forms, and comparing leading terms as x tends to infinity -- is not covered by CorbettMaths, Dr Austin Maths, Maths Genie, or Physics & Maths Tutor, none of which teach limits as a standalone GCSE/pre-calculus topic; consistent with sibling LO lo-gradient-of-tangents-on-curves' own prior 'no good match found' note)
Terminology
- limit
- tends to
- approaches
- direct substitution
- factorise
- common factor
- indeterminate form
- difference quotient
- infinity
- leading term
- limit does not exist
- gradient of tangent
Task goals
- Evaluate the limit of a polynomial as x tends to a given value by substituting the value directly into the expression.
- Recognise that direct substitution producing 0/0 is an indeterminate form, and factorise the numerator so the common factor cancels with the denominator before substituting again.
- Evaluate the limit of a simple rational function as x tends to infinity, using the fact that a constant over a power of x tends to 0 and comparing the leading terms of the numerator and denominator.
- Apply factor-and-cancel to rational functions where both the numerator and the denominator factorise, including difference-of-squares, difference-of-cubes, and repeated-factor cases.
- Simplify and evaluate a difference-quotient limit as h tends to 0 for a quadratic expression by expanding the bracket, cancelling the common factor of h, then letting h tend to 0.
- Apply the same difference-quotient process to cubic and quartic expressions, including ones written in other variables such as a and b, and to a function given in the form [f(x + h) - f(x)]/h.
- Explain what it means for a limit to exist at a point x = a, and give an example of a function whose limit does not exist there.
- Find an unknown constant in a rational expression given the value of its limit, and set out a step-by-step argument showing that the limit of [(x + h)^n - x^n]/h as h tends to 0 equals n times x^(n-1) for a given value of n.
Key skills
- Evaluate limit by direct substitution
- Recognise indeterminate form
- Factorise and cancel to evaluate a limit
- Evaluate limit as x tends to infinity
- Compare leading terms of numerator and denominator
- Evaluate a difference quotient limit
- Explain whether a limit exists
Quick stats
- 81 total questions
- 3 difficulty levels
- Answers included
