Explain the connection between infinite limits and vertical asymptotes.

If the limit of a function as $x$ approaches $a$ is infinite, then $x=a$ is a vertical asymptote.

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Explain the connection between infinite limits and vertical asymptotes.

If the limit of a function as $x$ approaches $a$ is infinite, then $x=a$ is a vertical asymptote.

Why are vertical asymptotes important in calculus?

They indicate points where a function is undefined and help visualize the function's behavior, especially concerning limits.

Why check both left-hand and right-hand limits at potential vertical asymptotes?

To understand how the function behaves on both sides of the asymptote and confirm its existence.

Describe the behavior of $f(x) = \frac{1}{x-a}$ near its vertical asymptote.

As $x$ approaches $a$ from the right, $f(x)$ approaches positive infinity. As $x$ approaches $a$ from the left, $f(x)$ approaches negative infinity.

Explain why $ln(x)$ has a vertical asymptote at $x=0$ .

Because $\lim_{x\to 0^+} ln(x) = -\infty$ , and $ln(x)$ is undefined for $x \le 0$ .

How does factoring help find vertical asymptotes?

Factoring can simplify rational functions, revealing removable discontinuities and true vertical asymptotes.

Explain the difference between a vertical asymptote and a removable discontinuity.

A vertical asymptote occurs where the function approaches infinity, while a removable discontinuity is a hole in the graph that can be 'removed' by redefining the function.

What does the sign of the infinite limit tell you about the function near a vertical asymptote?

A positive infinite limit means the function increases without bound, while a negative infinite limit means it decreases without bound.

Why is it important to simplify a rational function before finding vertical asymptotes?

Simplifying can reveal common factors in the numerator and denominator, which indicate removable discontinuities rather than vertical asymptotes.

Explain how the graph of a function behaves as it approaches a vertical asymptote.

The graph will approach the vertical asymptote very closely, with the y-values either increasing without bound (approaching positive infinity) or decreasing without bound (approaching negative infinity).

What is a vertical asymptote?

A vertical line $x=a$ that a function approaches but never touches; function is unbounded at $x=a$ .

What is a discontinuity?

A point where a function is undefined or behaves erratically.

What are infinite limits?

Limits that evaluate to either positive or negative infinity.

What does it mean for a function to be unbounded?

The function's value grows or shrinks without any bound, approaching infinity or negative infinity.

What is the relationship between vertical asymptotes and discontinuities?

A vertical asymptote is a type of discontinuity where the function is undefined.

Define $\lim_{x\to a} f(x) = \infty$ .

As $x$ approaches $a$ , the value of $f(x)$ increases without bound.

Define $\lim_{x\to a} f(x) = -\infty$ .

As $x$ approaches $a$ , the value of $f(x)$ decreases without bound.

What is a one-sided limit?

A limit that considers the function's behavior as x approaches a value from either the left or the right.

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

The limit of $f(x)$ as $x$ approaches $a$ from the right.

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

The limit of $f(x)$ as $x$ approaches $a$ from the left.

What does a vertical asymptote on the graph of $f(x)$ indicate about $f'(x)$ ?

It suggests that $f'(x)$ may also have a vertical asymptote or be unbounded near that x-value.

How can you identify a vertical asymptote from a graph?

Look for a vertical line that the function approaches but never crosses; the function's value tends towards $\pm \infty$ near this line.

What does the graph of $f(x) = \frac{1}{x}$ tell us about its limits near $x=0$ ?

It shows that $\lim_{x\to 0^+} f(x) = \infty$ and $\lim_{x\to 0^-} f(x) = -\infty$ , indicating a vertical asymptote at $x=0$ .

How does the graph of $ln(x)$ confirm its vertical asymptote?

The graph approaches the y-axis ( $x=0$ ) very closely as $x$ approaches 0 from the right, and the function's value goes to negative infinity.

How can you determine the sign of the infinite limit from a graph near a vertical asymptote?

If the graph goes up towards the asymptote, the limit is positive infinity; if it goes down, the limit is negative infinity.

If a graph has a vertical asymptote at x=a, what does that imply about the domain of the function?

The function is not defined at x=a, so x=a is not in the domain.

How does the steepness of a graph near a vertical asymptote relate to the limit?

The steeper the graph gets as it approaches the asymptote, the faster the function is approaching infinity (either positive or negative).

What graphical feature indicates a removable discontinuity rather than a vertical asymptote?

A hole in the graph, indicating a point where the function is undefined but could be defined to make the function continuous.

How can you graphically distinguish between $\lim_{x\to a^+} f(x) = \infty$ and $\lim_{x\to a^-} f(x) = \infty$ ?

Both limits indicate the graph goes up near x=a. The first approaches from the right, the second from the left.

What does the graph of a function with a vertical asymptote at x=a look like near that point?

The graph will approach the vertical line x=a very closely, with the y-values either increasing or decreasing without bound.