Limits notes Tiny Bit

Limits and Continuity

• Definition of Limits
    - Left-hand limit: The limit of a function as it approaches a point from the left side.
    - Right-hand limit: The limit of a function as it approaches a point from the right side.
    - Limit exists at a point if both one-sided limits exist and are equal.

• Graphical Explanation
    - Consider the case when evaluating the limit as $x$ approaches $0$.
      - Limit from the left:
        - $ext{lim}<em>{x o 0^-} f(x) = 2$       - The graph approaches $2$ from the left side.       - Note: Approaches does not imply that the value is equal to $2$ at $x=0$.     - Limit from the right:       - $ext{lim}</em>{x o 0^+} f(x) = 1$
        - The graph approaches $1$ from the right side.

• Limit Existence Conditions
    - For the limit as $x$ approaches $0$ to exist:
      - Both the left-hand limit and right-hand limit must be equal.
        - Since $2 <br />\neq 1$, the limit at $x = 0$ does not exist.

• Other Limit Evaluations
    - Limit as $x$ approaches $2$:
      - $ext{lim}<em>{x o 2} f(x) = 0$ (exists because both sides approach $0$)   - Limit as $x$ approaches $4$:     - $ext{lim}</em>{x o 4} f(x) = 2$ (exists)
    - Limit as $x$ approaches $6$:
      - $ext{lim}_{x o 6} f(x) = 0$ (exists)

• Key Realization
    - The critical limit is at $x = 0$ where limits from both sides are not equal, resulting in a non-existent limit.

Concavity and Points of Inflection

• Understanding Concavity
    - Concave down: Graph resembles a frown.
    - Concave up: Graph resembles a smile.

• Identifying Points of Inflection
    - Location where a function changes from concave up to concave down or vice versa.
    - For instance:
      - If a graph starts concave down and switches to concave up, there is a point of inflection at that transition.
      - Another point of inflection occurs when the graph switches back to concave down.

• Real-world Example
    - Discussing aspects of volume related to water:
      - Water volume is constantly increasing, which implies that as water continues to flow, the overall volume must increase over time.
      - The efficiency of this flow also relates back to the concavity discussion, as positive concavity in growth trends indicates increasing volume in scenarios such as filling a container.