Evaluating Limits

Reading

• Sections 2.5

Practice problems

• Section 2.5: 5, 7, 11, 13, 21, 24, 25
• To turn in (together with 2.6): 2.5 16, 30

Notes

Algebraic Evaluation of Limits

• Many limits cannot be evaluated by substitution.
• Most typical case is a “0 over 0”.
• Example: \(\displaystyle \lim_{x\to 2}\frac{x^2-4}{x-2}\)
• Solution: Perform algebraic manipulation to the function, without changing the value but eliminating the problematic part.
• In this example: \(\frac{x^2-4}{x-2} = \frac{(x-2)(x+2)}{x-2} = x+2\)

• We were able to eliminate the term \(x-2\), which was the one causing the zeros.

  When dealing with limits that cannot be evaluated directly:

  • Perform algebraic transformations until the problematic terms go away.
  • Evaluate limit of resulting expression by substitution/plugging in.

• Other examples:
  • \(\displaystyle\lim_{x\to 4}\frac{\sqrt{x} - 2}{x-4}\). Use “conjugate”.
  • \(\displaystyle\lim_{x\to 1}\left(\frac{1}{x-1} - \frac{2}{x^2-1}\right)\) (\(\infty - \infty\) form). Make common denominators.
  • \(\displaystyle\lim_{x\to \frac{\pi}{2}}\frac{\tan x}{\sec x}\). Write in terms of \(\sin\), \(\cos\).