MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
Let f(x)=x−3x2−9, and consider the behavior of f near x=3.
(a) When x is replaced by 3 in both the numerator and the denominator, what numerical expression is obtained? Identify the resulting form.
(b) Explain, without performing any additional algebraic manipulation, why the value found in part (a) alone is insufficient to determine x→3limf(x). Your explanation should reference the definition of a limit.
(c) Describe how rewriting f(x) in an equivalent form for all x=3 can resolve the indeterminate form identified in part (a). State what feature of the graph of y=f(x) at x=3 corresponds to the limit once it is determined.