Introduction to Limits and Function Behavior

Evaluation of Functions and Introduction to Limits

Evaluating a function involves determining the output value for a specific given input. For example, given the function $f(x) = \sqrt{x - 5}$, several points can be calculated:

• If $x = 8$, then $f(8) = \sqrt{8 - 5} = \sqrt{3}$. This is approximately $1.713$.

• If $x = 9$, then $f(9) = \sqrt{9 - 5} = \sqrt{4} = 2$.

• If $x = 6$, then $f(6) = \sqrt{6 - 5} = \sqrt{1} = 1$. This is exactly $1$, with a related approximation of $1.414$ mentioned in the context of other square roots.

• If $x = 10$, then $f(10) = \sqrt{10 - 5} = \sqrt{5}$, which is approximately $2.236$.

Evaluating a function specifically means that if $x$ is equal to a number, such as $x = 8$, you find $f(8)$ by "plugging it in." In contrast, a limit asks how a function behaves as its input changes or approaches a specific value.

The Conceptual Definition of a Limit

A limit examines the value that a function's output ($y$) approaches as the input ($x$) gets closer and closer to a specific point. This is distinct from the actual value of the function at that point.

• Approaching from the Left: As $x$ values increase from $6$ to $7$ to $8$, the corresponding $y$ values move from $1$ to $1.4$ to $1.7$. The value appears to be getting larger as $x$ approaches $8$.

• Approaching from the Right: Looking at $x = 10$ ($y \approx 2.236$) and moving toward $x = 8$, the $y$ values decrease as they approach the target value.

• Core Question: The focus is not necessarily what the function is equal to at a point, but what it looks like it should be equal to based on the surrounding values. This is the central idea of a limit in calculus.

• Principle of Reach: It does not matter if $y$ ever actually reaches the number; the limit is strictly about what value the function approaches.

Graphical Analysis of Limits and Function Values

Consider a function graph with various points and a specific hole at $x = 4$:

• At $x = 2$: The function value $f(2)$ is $1.5$.

• At $x = 3$: The function value $f(3)$ is $2$.

• At $x = 5$: The function value $f(5)$ is $3$.

• At $x = 6$: The function value $f(6)$ is $3.5$.

• At $x = 4$:

  • Evaluation: $f(4)$ does not exist (DNE). There is an open circle on the graph, meaning there is no defined $y$ value at exactly $x = 4$. If entered into a calculator, this would result in an error.

  • The Limit: As $x \rightarrow 4$, the graph approaches the value of $2.5$. Even though there is no point at $(4, 2.5)$, the points leading up to it (e.g., at $x = 3.1, 3.2$) and the points coming back from the right (e.g., from $x = 6$ down toward $4$) all converge toward the $y$ value of $2.5$.

Methods for Observing Limits

Limits can be analyzed through three primary lenses:

1. Numerically: Looking at sequences of numbers or tables of values.

2. Graphically: Observing the behavior of the line/curve on a coordinate plane.

3. Algebraically: Using mathematical formulas and manipulation.

Notation and One-Sided Limits

Specific notation is used to indicate which direction the input is approaching from:

• Left-hand Limit: Written as $x \rightarrow a^-$. The minus sign (placed like an exponent) indicates approaching from the left side or the negative infinity side.

• Right-hand Limit: Written as $x \rightarrow a^+$. The plus sign indicates approaching from the right side or the positive infinity side.

Numerical Examples of Sequences

• Approaching $2.25$ from the left: A sequence like $2.24, 2.249, 2.2499, 2.24999$ shows the values getting increasingly closer to $2.25$ from the smaller (left) side.

• Approaching $5.5$ from the right: A sequence like $5.51, 5.501, 5.5001$ indicates approaching $5.5$ from the larger (right) side.

• Approaching $1$: Sequences can approach a whole number in small increments, such as the ratios $\frac{7}{8}, \frac{15}{16}, \frac{31}{32}, \frac{63}{64}$, which represent a higher and higher percentage of $1$.

Existence of a Limit

For a limit to exist at a point $c$, the function must approach the same value from both the left and the right sides.

• Formal Requirement: $\lim_{x \rightarrow a^-} f(x) = \lim_{x \rightarrow a^+} f(x)$. If these two one-sided limits are equal to the same value $L$, then $\lim_{x \rightarrow a} f(x) = L$.

• Case of Non-Existence (Asymptotes): Consider $f(x) = \frac{1}{x - 6}$.

  • There is a vertical asymptote at $x = 6$.

  • As $x \rightarrow 6$ from the left, the function increases toward positive infinity ($+\infty$).

  • As $x \rightarrow 6$ from the right, the function decreases toward negative infinity ($-\infty$).

  • Since they go to different places (and infinity is a destination, not a number), the limit does not exist.

  • Domain of $f(x) = \frac{1}{x - 6}$: The function is defined on $(-\infty, 6) \cup (6, \infty)$.

Limits with Discontinuities

A limit can exist even if there is a hole or the point is defined elsewhere. Specifically:

• Example: A graph has a hole at $(4, 5)$ but a solid dot exists at $(4, 3)$.

  • $f(4) = 3$.

  • $\lim_{x \rightarrow 4} f(x) = 5$.

  • The limit cares only about what value the function approaches ($y = 5$), not where the actual point is ($y = 3$).

Indeterminate Forms

When evaluating a limit algebraically, such as $\lim_{x \rightarrow 3} \frac{x^2 - 9}{x - 3}$, direct substitution may fail:

• Substitution results in $f(3) = \frac{3^2 - 9}{3 - 3} = \frac{0}{0}$.

• The form $\frac{0}{0}$ is an indeterminate form. It is never a final answer.

• This result acts as a trigger to use a different method, such as numerical approximation ($x = 2.9, 2.99, 2.999$ and $x = 3.1, 3.01, 3.001$) or algebraic simplification. In this specific case, the limit as $x \rightarrow 3$ is actually $6$.

Questions & Discussion

Question: Just to clarify, you're saying that whenever you have the negative sign behind [the number], that basically means you're going from left to right, and then your positive sign is you're going right to left?

Response: That is exactly correct. The sign (placed as a superscript) indicates which infinity side you are coming from. The negative sign means you are coming from the negative infinity side (left to right), and the positive sign means you are coming from the positive infinity side (right to left). Even if the numbers themselves are positive (like approaching the number $4$ through $1, 2, 3$), the direction dictates the sign used in the limit notation.

Question: So, essentially, your limit is your y-value?

Response: Yes, the limit represents a $y$-value. However, it is important to distinguish that it is the $y$-value the function approaches, which may or may not be the actual value of the function at that specific $x$.