TL6 Algebraic Limits

Summary So Far¶

Now that we’ve introduced the idea of limits graphically, we are ready to see how we can calculate these limits algebraically. Remember: we’re building the terminology and techniques needed to construct the tangent line precisely.

Direct Evaluation¶

Polynomials¶

Calculating the limit of a polynomial is easy, as they do not have any jumps, breaks, or holes. Graphically, this means the point we approach is actually a point on the curve.

Rational Functions¶

Even though rational functions are made up of polynomials, the fact that there is division, allows for the possibility of division by 0. This can cause issues with the graph and ultimately complicate the limit calculation. Graphically, division by 0 can cause jumps, breaks, or holes in the graph of the function.

Indeterminate Form¶

The most common limit issue we will encounter is what we call a $\dfrac{0}{0}$ indeterminate form. The strategy for dealing with this is:

1. Try factoring both the numerator and denominator.

2. Look for a common factor to cancel.

3. Try direct evaluation again.

Primary Examples¶

Homework¶

Write out each question as well as your solution on the homework page template.

• Make sure you leave a good amount of blank space between each question.

• Write down the question as well as your solution.