Summary of Calculus 1 Review - Basic Introduction

Main Ideas and Concepts:

• Understanding Limits:

  A limit examines the behavior of a function as the input approaches a specific value. The limit is denoted as limx → a f(x).

• Direct Substitution:

  The first approach to finding a limit is Direct Substitution. If substituting the value leads to an undefined result (like 0/0), alternative methods must be used.

• Evaluating Limits with Algebraic Techniques:

  If Direct Substitution results in an undefined value, algebraic simplification (such as factoring) can help. For example, (x2 - 4)/(x - 2) can be factored to (x - 2)(x + 2)/(x - 2), allowing for cancellation and Direct Substitution.

• Complex Fractions:

  For Complex Fractions, multiplying by the common denominator can simplify the expression.

• Graphical Interpretation:

  Limits can also be evaluated graphically by analyzing the function's behavior from the left and right of a specific point.

• Types of Discontinuities:
  • Jump Discontinuity: Occurs when the left and right Limits at a point do not match.
  • Removable Discontinuity (Hole): Exists when the limit exists but does not equal the function value at that point.
  • Infinite Discontinuity: Occurs when the limit approaches infinity.

Methodologies:

• Finding Limits via Direct Substitution: Check if Direct Substitution yields a defined value.
• Using Algebraic Techniques:
  • Factor expressions when Direct Substitution results in 0/0.
  • Simplify Complex Fractions by multiplying by the common denominator.
  • For square roots in fractions, multiply by the conjugate.
• Graphical Evaluation:
  • Determine one-sided Limits (from the left and right).
  • Check if the Limits from both sides match to confirm the existence of a limit.

Example Problems:

• Limit as x → 3 of x2 + 5x - 4: Direct Substitution yields 20.
• Limit as x → 3 of (x2 - 8x + 15)/(x - 3): Factor to find limit equals -2 after canceling terms.
• Limit of a Complex Fraction: Multiply by the common denominator and simplify.
• Graphical Limits: Analyze values approaching from both sides and determine continuity or discontinuity.

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