What does the Existence of a Limit Theorem state?

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

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What does the Existence of a Limit Theorem state?

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

How do you estimate $\lim_{x \to a} f(x)$ using a table?

Create a table with x-values close to 'a' from both sides. 2. Calculate the corresponding f(x) values. 3. Observe the trend of f(x) as x approaches 'a'. 4. If f(x) approaches the same value from both sides, that's the estimated limit.

How do you determine if $\lim_{x \to a} f(x)$ exists using a table?

Evaluate f(x) for x-values approaching 'a' from both sides. 2. Check if f(x) approaches the same value from both sides. 3. If the values are the same, the limit exists; otherwise, it does not.

How do you handle an indeterminate form when evaluating a limit?

Recognize the indeterminate form (e.g., $\frac{0}{0}$ ). 2. Use a table to estimate the limit by choosing x-values close to the target value. 3. Observe the trend of the y-values to estimate the limit.

How do you create a table to estimate $\lim_{x \to 2} (x^2 -1)$ ?

Choose x-values close to 2 (e.g., 1.9, 1.99, 2.01, 2.1). 2. Calculate the corresponding y-values (e.g., (1.9)^2 - 1 = 2.61). 3. Observe the trend of the y-values.

How do you determine the one-sided limits from a table?

Focus on x-values approaching 'a' from the left (for the left-hand limit) or the right (for the right-hand limit). 2. Observe the trend of the corresponding y-values to estimate the one-sided limit.

How do you estimate $\lim_{x \to 0} \frac{\sin(x)}{x}$ using a table?

Create a table with x-values close to 0 (e.g., -0.1, -0.01, 0.01, 0.1). 2. Calculate the corresponding $\frac{\sin(x)}{x}$ values. 3. Observe the trend as x approaches 0.

How do you estimate $\lim_{x \to 3} f(x)$ using a table, given $f(x) = \frac{x^2 - 9}{x - 3}$ ?

Choose x-values close to 3 (e.g., 2.9, 2.99, 3.01, 3.1). 2. Calculate the corresponding $f(x)$ values. 3. Observe the trend as $x$ approaches 3.

How can you use a table to estimate $\lim_{x \to 1} \frac{x - 1}{\sqrt{x} - 1}$ ?

Create a table with x-values close to 1 (e.g., 0.9, 0.99, 1.01, 1.1). 2. Calculate the corresponding $f(x)$ values. 3. Observe the trend as $x$ approaches 1.

How do you estimate $\lim_{x \to 4} \frac{x - 4}{x^2 - 3x - 4}$ using a table?

Create a table with x-values near 4 (e.g., 3.9, 3.99, 4.01, 4.1). 2. Calculate the corresponding y-values. 3. Observe the trend of the y-values as x approaches 4.

How do you use a table to estimate $\lim_{x \to 2} f(x)$ , if $f(x)$ is only defined for $x > 2$ ?

Create a table with x-values approaching 2 from the right (e.g., 2.01, 2.001, 2.0001). 2. Calculate the corresponding f(x) values. 3. Observe the trend as x approaches 2 from the right.

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

All Flashcards

What does the Existence of a Limit Theorem state?

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

How do you estimate $\lim_{x \to a} f(x)$ using a table?

Create a table with x-values close to 'a' from both sides. 2. Calculate the corresponding f(x) values. 3. Observe the trend of f(x) as x approaches 'a'. 4. If f(x) approaches the same value from both sides, that's the estimated limit.

How do you determine if $\lim_{x \to a} f(x)$ exists using a table?

Evaluate f(x) for x-values approaching 'a' from both sides. 2. Check if f(x) approaches the same value from both sides. 3. If the values are the same, the limit exists; otherwise, it does not.

How do you handle an indeterminate form when evaluating a limit?

Recognize the indeterminate form (e.g., $\frac{0}{0}$ ). 2. Use a table to estimate the limit by choosing x-values close to the target value. 3. Observe the trend of the y-values to estimate the limit.

How do you create a table to estimate $\lim_{x \to 2} (x^2 -1)$ ?

Choose x-values close to 2 (e.g., 1.9, 1.99, 2.01, 2.1). 2. Calculate the corresponding y-values (e.g., (1.9)^2 - 1 = 2.61). 3. Observe the trend of the y-values.

How do you determine the one-sided limits from a table?

Focus on x-values approaching 'a' from the left (for the left-hand limit) or the right (for the right-hand limit). 2. Observe the trend of the corresponding y-values to estimate the one-sided limit.

How do you estimate $\lim_{x \to 0} \frac{\sin(x)}{x}$ using a table?

Create a table with x-values close to 0 (e.g., -0.1, -0.01, 0.01, 0.1). 2. Calculate the corresponding $\frac{\sin(x)}{x}$ values. 3. Observe the trend as x approaches 0.

How do you estimate $\lim_{x \to 3} f(x)$ using a table, given $f(x) = \frac{x^2 - 9}{x - 3}$ ?

Choose x-values close to 3 (e.g., 2.9, 2.99, 3.01, 3.1). 2. Calculate the corresponding $f(x)$ values. 3. Observe the trend as $x$ approaches 3.

How can you use a table to estimate $\lim_{x \to 1} \frac{x - 1}{\sqrt{x} - 1}$ ?

Create a table with x-values close to 1 (e.g., 0.9, 0.99, 1.01, 1.1). 2. Calculate the corresponding $f(x)$ values. 3. Observe the trend as $x$ approaches 1.

How do you estimate $\lim_{x \to 4} \frac{x - 4}{x^2 - 3x - 4}$ using a table?

Create a table with x-values near 4 (e.g., 3.9, 3.99, 4.01, 4.1). 2. Calculate the corresponding y-values. 3. Observe the trend of the y-values as x approaches 4.

How do you use a table to estimate $\lim_{x \to 2} f(x)$ , if $f(x)$ is only defined for $x > 2$ ?

Create a table with x-values approaching 2 from the right (e.g., 2.01, 2.001, 2.0001). 2. Calculate the corresponding f(x) values. 3. Observe the trend as x approaches 2 from the right.

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