Calculus AB - Unit 1: Limits and Continuity

Key Concepts

• Limits
  • Definition: The value that a function ( f(x) ) approaches as ( x ) approaches a certain point.
  • Example: For ( y = 3x ), ( \lim_{{x \to 1}} f(x) = 3 ).
  • Limits focus on approaching a point, not necessarily reaching it.
  • Piecewise functions may have limits even if the point does not exist.
  • One-sided limits:
    • Notated by a plus (from the right) or minus (from the left) sign next to the approaching value.
    • If left and right limits differ, the overall limit does not exist._

Operations with Limits

• Adding or multiplying limits follows intuitive rules.
• Simple limits (e.g., ( x^2 - 3 )) can be solved by substitution.

Special Cases

• Division by zero:
  • No zero over zero: Plug in values directly.
  • Zero over zero: Simplify by canceling factors or multiplying by conjugates.
• Trigonometric Limits:
  • Use identities where possible.
  • Special cases: ( \sin 5x / x ), use manipulations like multiplying by ( 5/5 ).
• The Squeeze Theorem:
  • Useful for bounding trigonometric limits. If ( f(x) \leq g(x) \leq h(x) ), same applies to limits.

Limits to Infinity

• Vertical Asymptotes: Limit equals infinity, indicating continuous increase or decrease.
• Horizontal Asymptotes: Limits as ( x \to \infty ), graph approaches a value but never reaches it.
• Solving:
  • Compare growth rates of numerator and denominator.
  • Use leading coefficients if growth rates are equal.
  • Larger growth on top results in infinity.

Continuity

• A function is continuous if it does not break.
• Types of discontinuities:
  • Point Discontinuity: Limit exists, but point does not.
  • Jump Discontinuity: Different limits from each side.
  • Infinite Discontinuity: Limits go to infinity.
• Continuity check involves ensuring limits from both sides equal the point.
• Piecewise functions can be made continuous by adjusting parameters.

Intermediate Value Theorem (IVT)

• States that a continuous function on an interval must take every value between ( f(a) ) and ( f(b) ).
• Example: For interval containing ( -1 ), function covers all values between end points.

Summary Notes

• Focus on understanding the basic rules of limits and continuity for the AP exam.
• Practice problem solving using these concepts.
• Further questions can be addressed in comments or future videos.