Vertical Asymptotes and Limits

What is the limit?

• 0

• 2

• +∞

• -∞

Answer explanation

Approaching 2 from the left means the denominator is slightly negative ⇒ the expression becomes very negative.

What is the limit?

• 0

• -∞

• +∞

• Undefined

Answer explanation

The graph of ln(x) approaches negative infinity as x approaches 0 from the right.

A function is undefined at every vertical asymptote.

• True
• False

Answer explanation

A function is always undefined at a vertical asymptote, because the values get infinitely large and do not settle on a single number.

Which of these functions has a vertical asymptote at x=3?

• 
• 
• 
• 

Answer explanation

Denominator = 0 at x = 3 in choice B ⇒ vertical asymptote.

What type of limit leads to a vertical asymptote?

• A finite limit

• An infinite limit

• A limit that does not exist

• A limit from only one side

Answer explanation

A vertical asymptote happens when the function grows without bound (positive or negative infinity) as x approaches a certain value.

So, the correct answer is:

B) An infinite limit leads to a vertical asymptote.

Which of the following statements is FALSE?

• A function can be undefined without having a vertical asymptote.

• Every undefined value must have a vertical asymptote.

• A function with a hole is undefined at that point.

• Vertical asymptotes occur when the limit is infinite.

Answer explanation

Not every undefined value leads to a vertical asymptote.

• - Some undefined points, like this one, just create a hole in the graph.

• - A vertical asymptote only happens if the limit becomes infinitely large.

Find the vertical asymptotes of f(x)

• x=-2 only

• x=0

• x=-2 and x=2

• No vertical asymptotes

Answer explanation

In the graph:

• The red and green dashed lines show the vertical asymptotes at x=2 and x=-2

As you can see, the function shoots off toward ∞ or −∞ near these lines.

What is the behavior of f(x) as x→0?

• Limit does not exist

• Limit = ∞

• Limit=0

• Function is defined at x=0

Answer explanation

From both sides, the value shoots up to infinity.

Which of the following is undefined at x=2 but does NOT have a vertical asymptote at x=2?

• 
• 
• 
• 

Answer explanation

The (x–2) cancels in (x-2)(x+1)/(x-2) ⇒ removable discontinuity (a hole).

Evaluate the limit

• 2

• ∞

• 0

• 1

Answer explanation