Settling zero over zero

Mathematics · Calculus & Analysis · ages 18-19

Name ______________________ Date ____________

What is the limit of (sin x)/x as x tends to 0?
Answer: ______________
Rank by growth as x goes to infinity, slowest first. Which ordering is right?
- ln x, x, x squared, e^x
- x, ln x, x squared, e^x
- e^x, x squared, x, ln x
- ln x, e^x, x, x squared
When may you use the rule?
- On 0 over 0 or infinity over infinity quotients
- On any quotient of differentiable functions
- On any product that tends to zero
Applying the rule means differentiating the whole quotient with the quotient rule.
Circle one: True False
What is the limit of x squared/e^x as x tends to infinity?
Answer: ______________
e to the x grows faster than any fixed power of x.
Circle one: True False
Apply the rule twice to find the limit of (1 - cos x)/x squared as x tends to 0.
- 1
- 0
- 0.5
- 2
e^x grows faster than any fixed power x^k. True or false?
Circle one: True False
What is the limit of x to the 100 over e to the x as x tends to infinity?
- It diverges to infinity
- 100
- It tends to 0
What is the limit of x^100/e^x as x tends to infinity?
- it diverges to infinity
- 100
- 1
- it tends to 0

Answer key

For grown-ups. Fold this page away before handing over the rest.

1 · 1. The classic limit: sine hugs the diagonal near zero.
ln x, x, x squared, e^x · ln x, x, x squared, e^x. Logs crawl, powers stride, exponentials fly.
On 0 over 0 or infinity over infinity quotients · Only the two indeterminate quotients qualify; products must be reshaped first.
False · You differentiate the top and the bottom separately, never the quotient as one piece.
0 · 0. Exponentials dominate every power, crushing the ratio to zero.
True · Repeated rounds leave the exponential standing over a constant, so the ratio dies to zero.
0.5 · 0.5. Two rounds give (cos x)/2, which tends to 1/2.
True · True. Repeated applications leave the exponential standing over a constant.
It tends to 0 · No fixed power survives against the exponential, however large the exponent.
it tends to 0 · It tends to 0. No fixed power survives against the exponential, however large the exponent.