Understanding Limits and Function Values

Understanding Limits and Function Values

Introduction to Limits

When discussing limits, we examine the behavior of a function as it approaches a specific independent variable value, regardless of whether the function is defined at that exact point. The focus is on the value the function gets 'close enough' to.

One-Sided Limits

Limits can be evaluated from either the left or the right side of a specific point. This distinction is crucial for determining the overall limit's existence.

Limit from the Left

• Notation: $\lim_{x \to a^-} f(x)$
• This examines values of $x$ that are slightly less than $a$, approaching $a$ from the negative direction on the number line. For instance, approaching zero from the left (denoted as $0^-$) means looking at values like $-0.1, -0.01, -0.001$.

Limit from the Right

• Notation: $\lim_{x \to a^+} f(x)$
• This examines values of $x$ that are slightly greater than $a$, approaching $a$ from the positive direction on the number line. For instance, approaching zero from the right (denoted as 0^+}) means looking at values like $0.1, 0.01, 0.001$.

Existence of a Limit

The definition of a limit states that a limit exists at a point $x=a$ if, and only if, the limit from the left and the limit from the right at that point are equal.

• If $\lim<em>{x \to a^-} f(x) = L$ and $\lim</em>{x \to a^+} f(x) = L$, then the limit exists and is equal to $L$ ($\lim_{x \to a} f(x) = L$).
• If $\lim<em>{x \to a^-} f(x) \neq \ ext{lim}</em>{x \to a^+} f(x)$, then the limit Does Not Exist (DNE) at $x=a$.

Example: Limit at $x=0$

Consider a function where we evaluate the limit as $x$ approaches $0$.

• Approaching from the left ($0^-$): As $x$ approaches $0$ from the left, the function approaches a value of $2$. So, $\lim_{x \to 0^-} f(x) = 2$.
• Approaching from the right (0^+}): As $x$ approaches $0$ from the right, the function also approaches a value of $2$. So, $\lim_{x \to 0^+} f(x) = 2$.
• Conclusion: Since the left-hand limit ($2$) and the right-hand limit ($2$) are the same, the overall limit at $x=0$ exists and is $2$. Therefore, $\lim_{x \to 0} f(x) = 2$.

Example: Limit at $x=1$

Let's analyze the limit of a different function or the same function at another point, $x=1$.

• Approaching from the left ($1^-$): As $x$ approaches $1$ from the left, the function goes to a specific point (e.g., a certain $y$-value).
• Approaching from the right (1^+}): As $x$ approaches $1$ from the right, the function goes to a different specific point (a different $y$-value) than when approaching from the left.
• Conclusion: Because the limit from the left and the limit from the right are not the same at $x=1$, the limit at $x=1$ Does Not Exist (DNE). This is a key differentiator when evaluating limits.

Function Value at a Point (f(x))

The function value at a specific point, denoted as $f(a)$, refers to the actual output of the function when the input is precisely $a$. This is distinct from a limit, which describes the function's behavior near a point.

• On a graph, the value of $f(a)$ is indicated by a filled dot at the point $(a, f(a))$.
• At $x=0$: If there is no filled dot anywhere directly above or below $x=0$, then $f(0)$ does not exist. This means the function is not defined at that exact point.
• At $x=1$: Even if a limit does not exist at $x=1$ (as in the example above), the function value $f(1)$ can still exist if there is a filled dot at $x=1$. This dot explicitly tells you the function's value at that precise input, independent of approaching behavior.
• The presence of a filled dot indicates where the function is at a point, whereas limits indicate where the function is going.

Continuity

The discussion implicitly touches upon continuity. A function is continuous at a point $x=a$ if three conditions are met:

1. The limit $\lim_{x \to a} f(x)$ exists.
2. The function value $f(a)$ exists.
3. $\lim_{x \to a} f(x) = f(a)$.

In our examples:

• At $x=0$: The limit exists, but if $f(0)$ does not exist (as implied by no filled dot), the function is not continuous at $x=0$.
• At $x=1$: The limit does not exist, so the function is not continuous at $x=1$ (even if $f(1)$ exists).

This highlights the difference between a function being defined at a point, and a limit existing at a point, both of which are components for a function to be continuous.