Limits

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.