Evaluating a limit of a rational function

Published on December 3, 2024

Calculus

To evaluate the limit, direct substitution results in an indeterminate form (0/0). Factoring the numerator reveals a common factor of (x-3), which cancels with the denominator. Substituting x=3 into the simplified expression yields a final answer of 58.

Question

What is the value of lim⁡x→3x4−9x2+4x−12(x−3)(4−x)?x→3lim(x−3)(4−x)x4−9x2+4x−12?

Answer

Here’s how to solve the limit problem:

1. Try Direct Substitution:

First, we try to substitute

directly into the expression:

Substituting

gives us:

This is an indeterminate form, so we need to simplify the expression.

2. Polynomial Long Division or Factoring:

Since we got the indeterminate form

, it suggests that

is a factor of the numerator. We can perform polynomial long division or try factoring to simplify. Let’s try factoring by grouping:

3. Simplify the Expression:

Now we can simplify the original expression:

4. Substitute :

Now, substitute

into the simplified expression:

Answer:

Therefore, the limit is 58.

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