Calculus - 2.2 - The Limit of a Function

Limits of Functions

Estimating Limits

• Graphical Estimation:
  • Observe the behavior of $y = f(x)$ as $x$ approaches a certain value (e.g., $x = 1$).
  • Estimate the limit $\lim_{x \to a} f(x)$ by looking at what y-value the graph approaches.
  • Example: If as $x$ gets close to 1, $y$ gets close to 2, then $\lim_{x \to 1} f(x) = 2$.
• Numerical Estimation:
  • Create a table of values for $f(x)$ for $x$ values close to the value of interest.
  • Examine the table to see what value $f(x)$ approaches as $x$ gets closer to the target.
  • Example: Estimate $\lim_{x \to 2} (x^2 + 1)$ by plugging in $x$ values close to 2 (e.g., 1.9, 1.99, 2.01, 2.1) and observing that $f(x)$ approaches 5.
• Important Note:
  • The limit of $f(x)$ as $x$ approaches $a$ is NOT necessarily the same as the value of $f(a)$.
  • $\lim_{x \to a} f(x)$ depends on the behavior of $f(x)$ near $x = a$, not necessarily at $x = a$.
  • It's entirely possible that $f(a)$ is undefined, but $\lim_{x \to a} f(x)$ still exists.
  • When finding $\lim_{x \to a} f(x)$, we should NEVER consider $x = a$.

Minimum Required Steps for Graphing Problems

1. State Equation(s): Explicitly state the equation(s) you are graphing (e.g., $y1 = x^2 + 1, y2 = …$).
2. Sketch the Graph: Draw a sketch of the graph.
3. State the Viewing Window: Specify the viewing window used (e.g., $[X<em>{min}, X</em>{max}, X<em>{scl}] \times [Y</em>{min}, Y<em>{max}, Y</em>{scl}]$).
4. Identify the Solution: Clearly indicate where on the graph you found the solution.
5. State the Final Answer: Provide the final answer to the problem.

Minimum Required Steps for Numerical Problems (Tables)

1. State Equation(s): Explicitly state the equation(s) you are using to generate the table (e.g., $y1 = …, y2 = …$).
2. Table of Values: Create a table including several points on both sides of the value you are approaching.
3. State the Final Answer: Provide the final answer to the problem.

Limit Definition

• We write $\lim_{x \to a} f(x) = L$ and say "the limit of $f(x)$ as $x$ approaches $a$, equals $L$ if we can make the values of the function $f(x)$ arbitrarily close to $L$ (as close to $L$ as we like) by taking $x$ sufficiently close to $a$ (on either side) but not equal to $a$.

Examples

• Example 1:

  • Consider a function $f(x)$ where $f(1) = -1$ and $\lim_{x \to 1} f(x) = 2$.
  • This illustrates that the limit as $x$ approaches 1 can be different from the function's value at $x = 1$.

• Example 2: $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$

  • This is a fundamental limit that is important to memorize.
  • The table of values shows that as $x$ approaches 0, $\frac{\sin(x)}{x}$ approaches 1.

• Example 3: Piecewise Function

• Example 4:

  • Given $f(x) = \begin{cases} \frac{x}{x}, &amp; x \neq 0 \ 1, &amp; x = 0 \end{cases}$
  • a) $\lim<em>{x \to 0^-} f(x) = \lim</em>{x \to 0^-} \frac{x}{x} = 1$
  • b) $\lim<em>{x \to 0^+} f(x) = \lim</em>{x \to 0^+} \frac{x}{x} = 1$
  • However, $\lim_{x \to 0} \frac{3}{x}$ DNE because the left and right limits aren't equal.
  • c) $f(0) = 1$

Finding x values within a range

• Given $f(x) = x^2 - x + 2$, find the values of x, where $f(x)$ is within 0.1 of 4.
• That is, find x such that 3.9 < f(x) < 4.1
• Graph $y1 = x^2 - x + 2$, $y2 = 3.9$ and $y3 = 4.1$
• The intersections occur at (1.966, 3.9) and (2.033, 4.1).

Pitfalls and Considerations

• Guessing the Limit:
  • Guessing limits from calculated values can be misleading.
  • Calculators and computers can sometimes give wrong answers.
  • Tables and graphs only provide estimates of the limit.
• Algebraic Methods:
  • Algebraic methods are more foolproof for finding limits.
    Example: Find the limit $\lim_{x \to 0} \sin(\frac{1}{x})$
  • The limit DNE because $\sin(\frac{1}{x})$ oscillates and does not approach a fixed number.

One-Sided Limits

• Left-Hand Limit:
  • $\lim_{x \to a^-} f(x) = L$ means the limit of $f(x)$ as $x$ approaches $a$ from the left equals $L$.
  • We can make the values of $f(x)$ arbitrarily close to $L$ by taking $x$ sufficiently close to $a$ and x < a.
• Right-Hand Limit:
  • $\lim_{x \to a^+} f(x) = L$ means the limit of $f(x)$ as $x$ approaches $a$ from the right equals $L$.
  • We can make the values of $f(x)$ arbitrarily close to $L$ by taking $x$ sufficiently close to $a$ and x > a.

*Heaviside function: $H = \begin{cases} 0 &amp; \text{if } t &lt; 0 \ 1 &amp; \text{if } t \geq 0 \end{cases}$
*Limit from the left: Approaches 0*
Limit from the right: approaches 1