The function h(x)= 1+ x -1 x is often encountered when investigating small-angle…
MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
The function
h(x)=x1+sinx−1
is often encountered when investigating small-angle phenomena.
(a) Explain why direct substitution of x=0 in h(x) leads to an indeterminate form.
(b) Describe, without performing the full computation, an algebraic manipulation that converts h(x) into a form where the limit as x→0 can be evaluated directly. Clearly state each algebraic or trigonometric identity you would invoke.
(c) Justify how the manipulation proposed in part (b) eliminates the indeterminacy and guarantees that the resulting limit is finite.