Concept preview

What you will be able to do

1. 01Identify the indeterminate form $\tfrac{0}{0}$ that appears when direct substitution fails, and explain why it does not mean the limit is undefined.
2. 02Evaluate limits by factoring and cancelling the common factor that causes the $\tfrac{0}{0}$ form, then substituting into the simplified expression.
3. 03Evaluate limits that require rationalizing a conjugate or combining a complex fraction over a common denominator before cancelling.

You reach for these tools the moment direct substitution gives $\tfrac{0}{0}$. That form is a genuine puzzle: the numerator wants to pull the fraction to $0$, while the denominator wants to blow it up, and the true limit is the outcome of that tug-of-war. Because the answer depends entirely on which factors are causing the zeros, no shortcut exists — you must rewrite the expression algebraically to see the resolution.