Problem

Let

Find conditions on coefficients a and e so that

Solution

To evaluate the limit of h(x) as x approaches 1, it is only necessary to evaluate h(x). The function h(x) can be evaluated easily at x = 1; it is equal to 7. To evaluate the limit of k(x) as x approaches 1, we have to concern ourselves with the possibility of division by zero. Consider the case first where e is not equal to –2. Then, for the two limits to be equal, we must have

Next consider the case e = –2. If e does equal –2, the denominator will equal 0 when x approaches 1, so we must be able to factor (–2x + 2) into the numerator ax² + 2x + 1 in order to evaluate the limit of k(x). For this to happen, we must be able to write the numerator as

This lets us solve for all the unknowns:

implies

implies

implies

Therefore, we have

But when we evaluate this limit, we get

a contradiction of our assumption that e = –2 and that h(x) could be factored. Thus, in order for the limits of h(x) and k(x) to be the same as x approaches 1, we must have the conditions