Some problems provide the opportunity for more than one mistake.

The Goal

Determine

The Mistakes

Find the mistakes:

1.

(Roll the mouse over the math to see a hint in red)

(The notation "l'H" above the equals sign indicates a step at which l'Hôpital's rule is claimed to be used)

2.

(Roll the mouse over the math to see a hint in red)

A Correct Solution

(Roll the mouse over the area above to see the corrections in blue)

Explanations

In the first mistake the student has misunderstood l'Hôpital's rule to apply to a product. L'Hôpital's rule can be used to find the limit of a quotient when both the numerator and denominator are differentiable on an open interval containing the point of interest (or on an infinite open interval if the limit is to be taken at ±∞) and when the limit has indeterminate form, that is, the numerator and denominator both approach zero in the limit or each approaches either ∞ or -∞. Thus the first step is to rearrange the limit expression into the form of an indeterminate quotient.

The second mistake shows one way to correctly rearrange the expression; the mistake comes only at the end of the calculation when the final rational expression is simplified incorrectly.