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What is a limit?

The value a function approaches as the input (x) gets closer to a specific value.

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What is a limit?

The value a function approaches as the input (x) gets closer to a specific value.

Define one-sided limit.

The value a function approaches as the input (x) gets closer to a specific value from either the left or the right.

What does $\lim_{x \to a^+} f(x) = L$ mean?

The limit of f(x) as x approaches 'a' from the right (values greater than 'a') is L.

What does $\lim_{x \to a^-} f(x) = L$ mean?

The limit of f(x) as x approaches 'a' from the left (values less than 'a') is L.

What is an indeterminate form?

An expression, like $\frac{0}{0}$ , where the limit cannot be determined by direct substitution.

What does it mean for x-values to be 'close' to a?

Numbers in close proximity to 'a' from both the left and right sides.

When does a limit exist?

A limit exists only if the left-hand and right-hand limits are equal.

What is direct substitution?

Evaluating a limit by plugging in the value that x approaches into the function.

What is a vertical asymptote?

A vertical line x = a where the function approaches infinity or negative infinity as x approaches a.

Define $f(x)$

A function that takes an input $x$ and returns an output

What does the Existence of a Limit Theorem state?

$\lim_{x \to a} f(x)$ exists if and only if $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$ .

Explain the concept of estimating limits from tables.

Choose x-values close to 'a' from both sides, calculate the corresponding y-values, and observe if the y-values approach a common value.

Explain the significance of one-sided limits in determining if a limit exists.

For a limit to exist, the left-hand limit and the right-hand limit must be equal. If they are not, the limit does not exist.

Why do we use tables to estimate limits?

Tables are used when direct substitution results in an indeterminate form, allowing us to observe the function's behavior near a specific point.

What does it mean if the y-values in a table do not approach the same value from both sides?

It suggests that the limit does not exist or that there might be a vertical asymptote at that point.

Explain how tables help estimate $\lim_{x \to a} f(x)$ when direct substitution fails.

By evaluating $f(x)$ for $x$ values close to $a$ from both sides, we can observe the trend of $f(x)$ and estimate the value it approaches.

Describe the relationship between one-sided limits and the existence of a two-sided limit.

A two-sided limit exists if and only if both one-sided limits exist and are equal to the same value.

Explain why knowing the value of $f(a)$ is not necessary when finding $\lim_{x \to a} f(x)$ .

The limit describes the behavior of $f(x)$ as $x$ approaches $a$ , not necessarily the value of $f(x)$ at $x = a$ .

Explain the process of using a table to determine if $\lim_{x \to a} f(x)$ exists.

Evaluate $f(x)$ for $x$ values approaching $a$ from both sides. If $f(x)$ approaches the same value from both sides, the limit exists.

Describe the importance of choosing appropriate x-values when estimating limits from tables.

Choosing x-values too far from $a$ may not accurately reflect the behavior of the function near $a$ , leading to an incorrect estimation.

Explain how a table can indicate that a limit does not exist.

If the values of $f(x)$ approach different values as $x$ approaches $a$ from the left and right, the limit does not exist.