This activity uses linearization of a function at a point as a substitution for the function in a limit such as sin(x) over (x) as x approaches 0. This can help de-mystify the limit of a ratio of 'competing' functions (provided x approaches a real number).
In particular, visualization of the tangent/linearization at a point, is made dynamic by TI-Nspire and produces a convincing argument that sin(x) over (x) as x approaches 0 is 1.
Before the Activity
Download the attached .pdf or .doc file and read through the activity.
Students should be able to determine the linearization of y = sin(x) at a point and should understand the Chain Rule. The activity may best be done as a prelude to L'Hospital's Rule.
During the Activity
During the activity it is important to emphasize that not all limits are as simple as applying the concept of substituting linearization. All linearizations have errors depending on where you are in the domain.
After the Activity
Students' understanding can easily be assessed through standard limit questions such as lim as x approaches zero of sin(7x) over 2 including justification.