basic continuity
What is the definition of a continuous function?
A function is continuous if there are no breaks, jumps, or holes in its graph.
What is a horizontal asymptote?
A horizontal line y=L that a function approaches as x→±∞
What does limx→∞ f(x) mean?
It describes what the function approaches as x gets infinitely large.
What is a vertical asymptote?
A vertical line x=a where the function grows toward ±∞
What type of asymptote does f(x)= 1/x have at x=0
A vertical asymptote at x=0
What are the three conditions needed for a function to be continuous at x=c
1) f(c) is defined, 2) lim x → f(x) exists, 3) lim x →f( x)= f(c)
How do you determine if a rational function has a horizontal asymptote?
Compare the degrees of the numerator and denominator.
What is the first step when solving a limit at infinity for a rational function?
Compare the degrees of the numerator and denominator.
How do you find vertical asymptotes of a rational function?
Set the denominator equal to zero and solve for x
True or False: Every rational function has both a horizontal and vertical asymptote
False—some functions have only one or neither.
True or False: A function with a hole is always discontinuous.
True, but it may be removable if the hole can be filled.
What happens if the degree of the numerator is greater than the denominator?
There is no horizontal asymptote; the function diverges.
What does it mean if a function’s limit at infinity equals a finite number?
The function has a horizontal asymptote at that value.
What must be true for a function to have a vertical asymptote at x=ax
The denominator must be zero, and the numerator must be nonzero at x=ax
What kind of asymptote does an exponential function like f(x)=2x have?
A horizontal asymptote at y=0
What type of discontinuity occurs if the limit exists but does not equal the function’s value
Removable discontinuity.
Can a function cross a horizontal asymptote?
Yes, a function can cross a horizontal asymptote but must approach it as x→±∞
If a rational function has a larger denominator degree, what is its limit at infinity?
The limit is zero.
What is the key difference between a hole and a vertical asymptote?
A hole occurs when a factor cancels; a vertical asymptote occurs when division by zero remains.
If a rational function has a larger numerator degree than denominator degree, what type of asymptote does it have?
No horizontal asymptote, but it may have a slant asymptote.
What is the first step in checking if a function is continuous at a specific point?
Check if the function is defined at that point.
If the numerator and denominator of a rational function have the same degree, how do you find the horizontal asymptote?
Divide the leading coefficients of the numerator and denominator.
Why do exponential functions like e^x grow infinitely as x→∞
Because exponentials grow faster than any polynomial.
What is the last step in confirming a vertical asymptote exists?
Check if the function approaches ∞ as x nears that value.
How can you tell if a rational function has a hole instead of a vertical asymptote?
.
If a factor cancels in the numerator and denominator, it's a hole, not a vertical asymptote.