Step 1: Direct Substitution Always try plugging in the value first. If you get a defined number, that's the answer.

Step 2: Identify the Indeterminate Form If substitution fails, identify which of the 7 forms you have. This determines your technique.

Step 3: Choose Technique Based on Form

For 0/0:

• Polynomial? Factorize.
• Square root present? Rationalize.
• Trig/exponential? Standard limits or Taylor.
• Complex? L'Hopital.

For infinity/infinity:

• Polynomial ratio? Divide by highest power.
• Otherwise? L'Hopital.

For 1^infinity:

• Apply e^(lim g*(f-1)) directly.

For 0 * infinity:

• Convert to 0/0 or inf/inf by flipping one factor.

For infinity - infinity:

• Common factor, rationalize, or combine fractions.

For 0^0 or $inf^0$:

• Take logarithm, evaluate lim g*ln(f), answer is e^(result).

Step 4: Verify Check your answer makes intuitive sense. For example, e^(negative) should be between 0 and 1.

Common Pitfalls:

• Don't use L'Hopital when a simpler method exists
• Don't forget to verify the indeterminate form before each L'Hopital application
• Don't confuse the quotient rule with separate differentiation
• For x -> negative infinity, remember sqrt($x^2$) = -x