All Flashcards

What is the notation for a limit?

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

What is the notation for a right-hand limit?

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

What is the notation for a left-hand limit?

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

What form indicates that direct substitution won't work?

$\frac{0}{0}$

What are the differences between using direct substitution and tables to find limits?

Direct Substitution: Simple, but fails for indeterminate forms. | Tables: Useful for indeterminate forms, but requires more computation and careful observation.

What are the differences between left-hand and right-hand limits?

Left-hand Limit: Approaching from values less than 'a'. | Right-hand Limit: Approaching from values greater than 'a'.

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.