By the Stolz-Cesaro theorem, this limit exists if
also exists, and the limits would be equal. The theorem requires that
be strictly monotone and divergent, which is the case since
.
You have
so we're left with computing
This can be done with the help of Stirling's approximation, which says that for large
,
. By this reasoning our limit is
Let's examine this limit in parts. First,
As
, this term approaches 1.
Next,
The term on the right approaches
, cancelling the
. So we're left with
Expand the numerator and denominator, and just examine the first few leading terms and their coefficients.
Divide through the numerator and denominator by
:
So you can see that, by comparison, we have
so this is the value of the limit.
Answer:
D
Step-by-step explanation:
42 x 2 = 84
150 - 84 = 66
66/2 = 33
Answer:
16x=1 x=1/16 or 0.0625
2/5x=1 x=5/2 or 2.5
Step-by-step explanation:
for 16x=1 divide both sides by 16, 1 divided by 16 is 1/16 or 0.0625
for 2/5x=1 multiply by the reciprocal of 2/5 which is 5/2. 5/2 times 1 is just 5/2 or 2.5.
Answer:
- <em>B. The grouping method of factoring trinomials involves rewriting the bx term into the factors that fit the particular trinomial, and factoring these four terms using grouping</em>
Explanation:
The description may be better explained by applying it to an example.
Example:
- the general form of a trinomial is a x² - bx - 30
- comparing with x² - x - 30 the <em>bx term </em>is - x
- then you must <em>rewrite the bx term, - x,</em> into two terms whose coefficients are factors of 30:
Two numbers which add up - 1 and multipled are - 30. Those numbers are - 6 and + 5, because -6 + 5 = - 1 and (-6) × (+5) = -30.
Hence, the two terms are -6x and 5x, and the expression rewritten is:
x² - 6x + 5x - 30
- <em>factor these four terms using grouping</em>:
(x² - 6x) + (5x - 30)
x(x - 6) + 5(x - 6)
(x - 6) (x + 5)
Hence, the factored trinomial is (x - 6) (x + 5)
Answer:
dilation 10 left and 3 down
Step-by-step explanation: