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Scrat [10]
3 years ago
9

Prove that the limit x tends to 1 (2x^4-6x^3+x^2+3)÷(x-1)=-8

Mathematics
1 answer:
elixir [45]3 years ago
5 0
First note that if x\neq1, you have

\dfrac{2x^4-6x^3+x^2+3}{x-1}=2x^3-4x^2-3x-3

Now, you're looking for \delta>0 such that for any \varepsilon>0, you have

|x-1|

Note that you can divide through the left side of the \varepsilon inequality by x-1 once more:

\dfrac{2x^3-4x^2-3x+5}{x-1}=2x^3-2x-5\implies 2x^3-4x^2-3x+5=(x-1)(2x^2-2x-5)

So it follows that you need to find an appropriate \delta that will guarantee

|(x-1)(2x^2-2x-5)|=|x-1||2x^2-2-5|

For the moment, let's fix \delta=1. Then by this assumption, we have

|x-1|

From this we get

\implies0
\implies0
\implies0
\implies-5
\implies1

where the upper bound is what we care about. With this assumption, we then get that

|x-1||2x^2-2-5|

which suggests that \delta can be taken to be either the smaller of 1 or \dfrac{\varepsilon}5, or \delta=\min\left\{\dfrac{\varepsilon}5,1\right\}, to guarantee that the function gets arbitrarily close to -8.
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3 years ago
2.1e-10 in standard form?
just olya [345]
<span>2.1e-10
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4 years ago
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What is the exponent of x in in the expression -4x^3y^2 + 6
liberstina [14]

Answer:

3

Step-by-step explanation:

- 4x³y² + 6

the x is raised to the power of 3 , that is exponent of x is 3

4 0
2 years ago
A survey is being conducted in a county where 62% of the voters are Democrats and 38% are Republican. (a) What is the probabilit
Inessa05 [86]

Answer:

0.3844 = 38.44% probability that two independently surveyed voters would both be Democrats

Step-by-step explanation:

For each voter, there are only two possible outcomes. Either the voter is a Democrat, or he is not. The probability of the voter being a Democrat is independent of other voters. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}

In which C_{n,x} is the number of different combinations of x objects from a set of n elements, given by the following formula.

C_{n,x} = \frac{n!}{x!(n-x)!}

And p is the probability of X happening.

62% of the voters are Democrats

This means that p = 0.62

(a) What is the probability that two independently surveyed voters would both be Democrats?

This is P(X = 2) when n = 2. So

P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}

P(X = 2) = C_{2,2}.(0.62)^{2}.(0.38)^{0} = 0.3844

0.3844 = 38.44% probability that two independently surveyed voters would both be Democrats

3 0
3 years ago
Members of the Granola Gang were popping popcorn over the campfire.
nordsb [41]

Answ

JOSUE

Step-by-step explanation:

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8 0
3 years ago
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