Comprehensive Study Notes on Limits and One-Sided Limits

Informal Definition and Intuition of Limits

• Wind Speed Analogy:

  • An observer starts at position $1$ with a fan located at position $4$.

  • To determine the wind speed produced by the fan as position $x$ gets close to $4$ (the exact location of the fan), wind speed is measured at positions increasingly close to $x = 4$.

  • Direct measurement at the exact position $x = 4$ cannot occur due to the risk of serious bodily injury.

  • Data collection begins at an $x$ value of $3$ and approaches positive $4$ (e.g., $x = 3.5$, $x = 3.9$, and continuing closer).

  • As $x$ approaches $4$, the corresponding function values or $y$-values approach $6\text{ miles per hour}$.

  • Mathematical notation for this relationship:

$\lim_{x \to 4} s(x) = 6$

• Informal Definition of a Limit:

  • As $x$ approaches $c$, the limit of $f(x)$ is $L$, written in standard mathematical notation as:

$\lim_{x \to c} f(x) = L$

• This definition holds if all values of $f(x)$ are close to $L$ for values of $x$ that are sufficiently close to, but explicitly not equal to, $c$.

• Values that are "sufficiently close" consist of numbers both less than $c$ (located to the left of $c$) and greater than $c$ (located to the right of $c$).

• Certain functions may not possess limits at specific values of $c$.

  • Independence from Function Value at $c$:

• The value of a limit is not affected by the actual value of the function at $x = c$.

• A function does not need to be defined at $x = c$ to possess a valid limit at $x = c$.

• Example evaluating $\lim_{x \to 1} f(x)$ where $f(x)$ is undefined at $x = 1$:

  • Using the "wall method," a vertical reference line is drawn through $x = 1$.

  • The wall is approached from both the left side and the right side to determine if the function approaches the same value.

  • Graphically, as $x$ approaches $1$ from the left side, $y$-values approach $-1$.

  • Numerically via a T-table, as $x$ values approach positive $1$ from the left, $y$-values approach $-1$.

  • Graphically, as $x$ approaches $1$ from the right side (indicated by a green arrow), $y$-values approach $-1$.

  • Numerically via a T-table, as $x$ values approach $1$ from values greater than $1$, $y$-values approach $-1$.

  • Because both left-sided and right-sided approaches yield $-1$, the limit exists:

$\lim_{x \to 1} f(x) = -1$

Approaches to Finding Limits

• Three General Methods for Finding Limits:

  • Numerical Approach: Utilizing a T-table (table of values) to observe functional output trends as input values approach $c$.

  • Graphical Approach: Visual analysis of a graph to trace function outputs as inputs approach $c$ from both directions.

  • Analytical Method: Applying algebraic manipulation or calculus techniques to calculate limits.

One-Sided Limits and Existence Theorem

• Definitions and Notation of One-Sided Limits:

  • Left-Sided Limit: Approaching $c$ strictly from values less than $c$ (from the left side). Designated by a negative sign superscript in the upper right-hand corner of $c$:

$\lim_{x \to c^-} f(x)$

• Right-Sided Limit: Approaching $c$ strictly from values greater than $c$ (from the right side). Designated by a positive sign superscript in the upper right-hand corner of $c$:

$\lim_{x \to c^+} f(x)$

• Limit Existence Theorem:

  • As $x$ approaches $c$, the limit of $f(x)$ equals $L$ if and only if the left-sided limit exists, the right-sided limit exists, and both one-sided limits are equal to $L$:

$\lim_{x \to c^-} f(x) = L$

$\lim_{x \to c^+} f(x) = L$

$\lim_{x \to c} f(x) = L$

• If the left-sided limit and right-sided limit approach different values, the overall limit does not exist.

  • Example of a Discontinuous Function at $x = 1$:

• Evaluating one-sided limits and overall limit as $x$ approaches positive $1$ using a vertical wall line at $x = 1$:

• Right-Sided Limit Evaluation:

  • Graphically, approaching positive $1$ along the green arrow from the right shows $y$-values approaching $-2$.

  • Numerically via the T-table, as $x$ approaches $1$ from the right side, $y$-values approach $-2$:

$\lim_{x \to 1^+} f(x) = -2$

• Left-Sided Limit Evaluation:

  • Graphically, approaching positive $1$ along the right arrow from the left shows $y$-values approaching positive $3$.

  • Numerically via the T-table, as $x$ approaches positive $1$ from values less than $1$, $y$-values approach positive $3$:

$\lim_{x \to 1^-} f(x) = 3$

• Overall Limit Evaluation:

  • Because the left-sided limit ($3$) and the right-sided limit ($-2$) are not equal, the overall limit does not exist:

$\lim_{x \to 1} f(x) \text{ does not exist}$

Infinite Limits and Asymptotic Behavior

• Example Evaluating Function $h(x)$ as $x$ Approaches $-2$:

  • Visualizing behavior using a yellow vertical line (wall) through $x = -2$:

  • Left-Sided Limit ($x \to -2^-$):

  • Approaching $-2$ along the red arrow from the left side causes the graph to go down indefinitely.

  • Function values approach negative infinity:

$\lim_{x \to -2^-} h(x) = -\infty$

• Right-Sided Limit ($x \to -2^+$):

  • Approaching $-2$ along the green arrow from the right side causes the graph to go up indefinitely.

  • Function values approach positive infinity:

$\lim_{x \to -2^+} h(x) = \infty$

• Overall Limit Evaluation:

  • Because the one-sided limits approach different behaviors ($-\infty$ vs $\infty$), the overall limit does not exist:

$\lim_{x \to -2} h(x) \text{ does not exist}$

• Conceptual Rule Concerning Infinity:

  • Infinity ($\infty$) and negative infinity ($-\infty$) do not represent real numerical values and do not exist as numbers.

  • Writing $\infty$ or $-\infty$ as a limit result is standard mathematical practice because it provides specific descriptive information regarding the unbounded behavior of a function's graph.