2.1 Concept and definition of limits

Limits are the foundation of calculus, describing how functions behave as inputs approach specific values. They're crucial for understanding derivatives and integrals, allowing us to analyze function behavior even at undefined points.

Evaluating limits involves various techniques, from creating tables and graphing to algebraic manipulation. These methods help determine if a limit exists and what its value is, providing insights into function behavior and continuity.

Understanding Limits

Concept and role of limits

Top images from around the web for Concept and role of limits

• 

  Limits at Infinity and Asymptotes · Calculus View original

  Is this image relevant?

• 

  Continuity · Calculus View original

  Is this image relevant?

• 

  The Limit of a Function · Calculus View original

  Is this image relevant?

• 

  Limits at Infinity and Asymptotes · Calculus View original

  Is this image relevant?

• 

  Continuity · Calculus View original

  Is this image relevant?

1 of 3

Top images from around the web for Concept and role of limits

• 

  Limits at Infinity and Asymptotes · Calculus View original

  Is this image relevant?

• 

  Continuity · Calculus View original

  Is this image relevant?

• 

  The Limit of a Function · Calculus View original

  Is this image relevant?

• 

  Limits at Infinity and Asymptotes · Calculus View original

  Is this image relevant?

• 

  Continuity · Calculus View original

  Is this image relevant?

1 of 3

• Limits characterize the behavior of a function as the input approaches a specific value without necessarily reaching it
  • Determine the output value a function gets close to as the input nears a certain point (0, $\pi$, $\infty$)
  • The function does not need to be defined at the point where the limit is being evaluated (removable discontinuity, hole in the graph)
• Limits form the foundation for key concepts in calculus and are used to rigorously define them
  • Derivatives represent the instantaneous rate of change of a function at a given point (velocity, slope of tangent line)
  • Integrals give the total area under a curve or the accumulation of a quantity (volume, work)
• Limits enable the analysis of functions at points where they may be undefined, have a vertical asymptote, or display unusual behavior (rational functions, piecewise functions)

Mathematical notation for limits

• For a function $f(x)$ and a real number $a$, the limit of $f(x)$ as $x$ approaches $a$ is denoted by:
  • $\lim_{x \to a} f(x) = L$
• This notation signifies that as $x$ gets arbitrarily close to $a$ from both sides, $f(x)$ approaches the value $L$
• The limit $L$ exists if the function approaches the same value from both the left and right sides of $a$
  • Left-hand limit: $\lim_{x \to a^-} f(x)$, where $x$ approaches $a$ from values less than $a$ (2.9, 2.99, 2.999 approaching 3)
  • Right-hand limit: $\lim_{x \to a^+} f(x)$, where $x$ approaches $a$ from values greater than $a$ (3.1, 3.01, 3.001 approaching 3)

Existence of limits

• A limit exists if the function approaches the same value from both the left and right sides of the point
• To determine the existence of a limit:
  1. Evaluate the left-hand and right-hand limits separately
  2. If both one-sided limits exist and are equal, the overall limit exists
  3. If either one-sided limit does not exist or they are not equal, the overall limit does not exist
• Limits may fail to exist for various reasons:
  • Vertical asymptote: the function grows without bound as $x$ approaches $a$ ($\frac{1}{x}$ as $x \to 0$)
  • Jump discontinuity: the function has different left and right limits at $a$ (signum function at 0)
  • Oscillating behavior: the function does not approach a single value as $x$ approaches $a$ (sin($\frac{1}{x}$) as $x \to 0$)

Methods for evaluating limits

• Tables: construct a table of $x$ and $f(x)$ values to identify the trend as $x$ approaches the point of interest
  • Select $x$ values progressively closer to the point from both sides (2.9, 2.99, 2.999 and 3.1, 3.01, 3.001 for a limit at 3)
  • If $f(x)$ values approach a single value, that value represents the limit
• Graphs: visually inspect the graph of the function near the point of interest
  • If the graph has a hole or a removable discontinuity at the point, the y-coordinate of the hole gives the limit ($\frac{x^2-1}{x-1}$ at $x=1$)
  • If the graph has a vertical asymptote or jump discontinuity, the limit does not exist ($\frac{1}{x}$ at $x=0$, step function at jump)
• Algebraic manipulation: apply limit laws and algebraic techniques to simplify the function and evaluate the limit
  • Direct substitution: if the function is continuous at the point, substitute the point directly into the function ($\lim_{x \to 2} (x^2+3x-5) = 2^2+3(2)-5 = 5$)
  • Factoring and canceling: factor the numerator and denominator to cancel common terms ($\lim_{x \to 3} \frac{x^2-9}{x-3} = \lim_{x \to 3} \frac{(x+3)(x-3)}{x-3} = \lim_{x \to 3} (x+3) = 6$)
  • Rationalization: multiply the numerator and denominator by the conjugate of the denominator to simplify the expression ($\lim_{x \to 1} \frac{\sqrt{x}-1}{x-1} = \lim_{x \to 1} \frac{(\sqrt{x}-1)(\sqrt{x}+1)}{(x-1)(\sqrt{x}+1)} = \lim_{x \to 1} \frac{x-1}{(x-1)(\sqrt{x}+1)} = \lim_{x \to 1} \frac{1}{\sqrt{x}+1} = \frac{1}{2}$)