Taming zero over zero

Mathematics · Calculus & Analysis · ages 18-19

Name ______________________ Date ____________

Substituting into a quotient gives 0 / 0, so the limit does not exist.
Circle one: True False
A function g satisfies minus x squared is less than or equal to g(x), and g(x) is less than or equal to x squared, for all x near 0. What is the limit of g(x) as x approaches 0?
- 0
- 1
- it cannot be determined
- it depends on the formula for g
Find the limit as x approaches 2 of (x squared minus 4) over (x minus 2).
- 0
- 2
- 4
- it does not exist
Suppose f heads to 5 and g heads to -1. What does f times g plus 3 head to?
- -8
- -2
- 2
Find the limit as x approaches 0 of sin(4x) over x.
Answer: ______________
Find the limit as x approaches 4 of (root x minus 2) over (x minus 4).
Answer: ______________
What is the limit of (root(x) - 2) / (x - 4) as x approaches 4? Give a decimal.
Answer: ______________
Find the limit as x approaches 0 of x squared times sin(1 over x).
Answer: ______________
Find the limit as x approaches 0 of tan(2x) over x.
Answer: ______________
What is the limit of x squared times sin(1 / x) as x approaches 0, and why?
- It does not exist, since the sine factor has no limit
- It is 0, since the product sits between -x squared and x squared
- It is 1, like sin(x) / x

Answer key

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

False · That form is undecided, not a verdict. Simplify and try again.
0 · When two outer functions squeeze g and both head to the same number, g is forced there too.
4 · 0/0 is a signal to factorise, not a final answer.
-2 · The product law gives -5, and the sum law adds 3.
4 · Rescale so the basic limit sin(u) over u to 1 applies, then multiply out the factor 4.
0.25 · Rationalising turns the surd difference into a factor you can cancel.
0.25 · Rationalising leaves 1 / (root(x) + 2), which is 1/4 at x = 4.
0 · The sine factor oscillates, but squeezing it between plus and minus x squared forces the limit.
2 · Split tan into sine over cosine and reuse the basic trig limit with u = 2x.
It is 0, since the product sits between -x squared and x squared · The sine wobble is trapped between bounds that both vanish.