What is the notation for the left-hand limit as x approaches a?

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

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What is the notation for the left-hand limit as x approaches a?

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

What is the notation for the right-hand limit as x approaches a?

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

How do you estimate a limit from a graph?

Visualize the point. 2. Trace along the graph from both sides. 3. Check if the one-sided limits match.

How do you determine if a limit DNE from a graph?

Check for vertical asymptotes. 2. Check for jump discontinuities. 3. Check for wild oscillations.

Given a graph, how do you find the left-hand limit at x=a?

Trace the graph from the left side of x=a. Determine the y-value the function approaches as x gets closer to a from the left.

Given a graph, how do you find the right-hand limit at x=a?

Trace the graph from the right side of x=a. Determine the y-value the function approaches as x gets closer to a from the right.

How do you evaluate $\lim_{x \to a} f(x)$ from a graph?

Determine $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ . If they are equal, that is the limit. Otherwise, the limit DNE.

How do you handle a graph with a hole at x=a when finding $\lim_{x \to a} f(x)$ ?

The limit can still exist even with a hole. Focus on what y-value the function approaches as x approaches 'a' from both sides, not the value at x=a.

How do you determine if a function is continuous at x=a from its graph?

Check if the limit exists at x=a, if f(a) is defined, and if $\lim_{x \to a} f(x) = f(a)$ .

How do you identify a jump discontinuity on a graph?

Look for a point where the graph 'jumps' from one y-value to another. The left and right limits will be different at this point.

How do you identify a vertical asymptote on a graph?

Look for a vertical line where the function approaches infinity (or negative infinity) as x approaches that line.

How do you deal with oscillations when estimating limits from a graph?

If the function oscillates wildly near a point, the limit likely does not exist. The function does not approach a single, finite value.

What is a limit?

The value a function approaches as the input approaches a specific point.

Define a left-hand limit.

The value a function approaches as x approaches a value from the left.

Define a right-hand limit.

The value a function approaches as x approaches a value from the right.

What does DNE mean in the context of limits?

Does Not Exist. The limit does not approach a single, finite value.

What is a vertical asymptote?

A vertical line that a function approaches but never touches; often indicates a limit DNE.

What is a jump discontinuity?

A point where the left-hand limit and right-hand limit exist but are not equal.

What is an oscillating function?

A function that fluctuates rapidly between two values as x approaches a certain point.

What is an unbounded function?

A function whose values increase or decrease without limit as x approaches a certain point.

What is the notation for the limit of f(x) as x approaches 'a'?

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

What does it mean for a function to be continuous at a point?

The limit exists at that point, the function is defined at that point, and the limit equals the function value.

All Flashcards

What is the notation for the left-hand limit as x approaches a?

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

What is the notation for the right-hand limit as x approaches a?

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

How do you estimate a limit from a graph?

Visualize the point. 2. Trace along the graph from both sides. 3. Check if the one-sided limits match.

How do you determine if a limit DNE from a graph?

Check for vertical asymptotes. 2. Check for jump discontinuities. 3. Check for wild oscillations.

Given a graph, how do you find the left-hand limit at x=a?

Trace the graph from the left side of x=a. Determine the y-value the function approaches as x gets closer to a from the left.

Given a graph, how do you find the right-hand limit at x=a?

Trace the graph from the right side of x=a. Determine the y-value the function approaches as x gets closer to a from the right.

How do you evaluate $\lim_{x \to a} f(x)$ from a graph?

Determine $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ . If they are equal, that is the limit. Otherwise, the limit DNE.

How do you handle a graph with a hole at x=a when finding $\lim_{x \to a} f(x)$ ?

The limit can still exist even with a hole. Focus on what y-value the function approaches as x approaches 'a' from both sides, not the value at x=a.

How do you determine if a function is continuous at x=a from its graph?

Check if the limit exists at x=a, if f(a) is defined, and if $\lim_{x \to a} f(x) = f(a)$ .

How do you identify a jump discontinuity on a graph?

Look for a point where the graph 'jumps' from one y-value to another. The left and right limits will be different at this point.

How do you identify a vertical asymptote on a graph?

Look for a vertical line where the function approaches infinity (or negative infinity) as x approaches that line.

How do you deal with oscillations when estimating limits from a graph?

If the function oscillates wildly near a point, the limit likely does not exist. The function does not approach a single, finite value.

What is a limit?

The value a function approaches as the input approaches a specific point.

Define a left-hand limit.

The value a function approaches as x approaches a value from the left.

Define a right-hand limit.

The value a function approaches as x approaches a value from the right.

What does DNE mean in the context of limits?

Does Not Exist. The limit does not approach a single, finite value.

What is a vertical asymptote?

A vertical line that a function approaches but never touches; often indicates a limit DNE.

What is a jump discontinuity?

A point where the left-hand limit and right-hand limit exist but are not equal.

What is an oscillating function?

A function that fluctuates rapidly between two values as x approaches a certain point.

What is an unbounded function?

A function whose values increase or decrease without limit as x approaches a certain point.

What is the notation for the limit of f(x) as x approaches 'a'?

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

What does it mean for a function to be continuous at a point?

The limit exists at that point, the function is defined at that point, and the limit equals the function value.