Overview
This lecture covers infinite limits, evaluating one-sided limits, direct substitution, vertical asymptotes, and how to determine holes in rational functions.
Infinite Limits and One-Sided Limits
- The limit of 1/x as x approaches zero from the left is negative infinity; from the right, it is positive infinity.
- If left- and right-hand limits do not match, the overall limit does not exist.
- The limit of 1/x² as x approaches zero from either side is positive infinity because the denominator, when squared, is always positive.
- The limit of 1/(x-2) as x approaches 2 from the left is negative infinity; from the right, it is positive infinity.
- The limit of 1/(x-1) as x approaches 1 from the left is negative infinity; from the right, it is positive infinity—so the limit does not exist.
- The limit of x³/(x-2)² as x approaches 2 from the right or left is positive infinity, as the denominator becomes very small positive values.
- For functions like ln(x) as x approaches zero from the right, the limit is negative infinity; from the left, the limit does not exist.
Evaluating Limits with Direct Substitution
- If plugging in the value does not give a zero denominator, use direct substitution.
- Example: Limit as x→5 of x²/(x²+25) is 25/50 = 1/2.
- If a factor can be canceled, do so before substituting, as in (x+3)/(x²+x−6).
Trigonometric and Logarithmic Limits
- The limit of tan(x) as x approaches π/2 does not exist because the left limit is positive infinity, the right limit is negative infinity.
- The limit of csc(x) as x approaches 0 from the right is positive infinity.
- The limit of sec(x) as x approaches π/2 does not exist; from the left, it is positive infinity, from the right, negative infinity.
Vertical Asymptotes and Holes
- Vertical asymptotes occur where the denominator equals zero and doesn't cancel with the numerator.
- To find vertical asymptotes, set denominator factors equal to zero and solve for x.
- If a factor cancels with the numerator, it creates a hole, not a vertical asymptote.
- Factors that result in complex solutions (e.g., x²+4=0) do not correspond to real asymptotes or holes.
Key Terms & Definitions
- Infinite limit — When a function increases or decreases without bound as x approaches a certain value.
- One-sided limit — The value a function approaches as x approaches a value from one side (left or right).
- Vertical asymptote — A vertical line x = a where a function grows without bound.
- Hole — A point where a factor cancels, resulting in the function being undefined at that x-value.
- Direct substitution — Plugging in the limit value directly into the function when possible.
Action Items / Next Steps
- Practice evaluating limits by substituting values close to the target x.
- Factor denominators completely when searching for asymptotes or holes.
- Complete assigned homework problems on evaluating limits and identifying vertical asymptotes.