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3 years ago

For what values of a and b does a - b = b - a?

Mathematics

1 answer:

Deffense [45]3 years ago

3 0

how can a+b=0?

is a+b=0 a single word followed by the question mark ? the sentence is not complete if “a+b=0” is the subject, it’s like: how can cat?

is the sentence rather: how can a+b be equal to 0 ( meaning zero) ? in that case, supposing we are talking of the usual sum of two elements of an additive group : how can the sum of a and b, written a+b, can be equal to zero, then this seems more like a question about what powers do have any two elements of a group to, by association through addition, to obtain a zero value. Like “how can my brother and me get one of these chocolates ?”

Now, in your question, maybe a and b are not any two elements of said group but elements well identified before,

suppose you know a is 1 and b is -1 for instance. Then how can 1–1=0? is a very good question : it marvels at the fact that the addition does what it is supposed to do. Lots of things don’t, and lots of people take it for granted that things do what they are supposed to de, like speeds add up, etc.

let’s suppose for example, a is 1 and b is 2.

Then your question is very justified : how can 1+2 be equal to 0 ? it’s a scandal, either the computer is broken, or my mind is , and if anywhere in the world 1+2 =0, there is something wrong with the world.

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Hope this helps!! - Maddy

(if this helps please make brainliest, rate, and thank so I know how I did!)

╰(*°▽°*)╯

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3 years ago

A) Evaluate the limit using the appropriate properties of limits. (If an answer does not exist, enter DNE.)

Gelneren [198K]

For purely rational functions, the general strategy is to compare the degrees of the numerator and denominator.

$\displaystyle \lim_{x\to\infty} \frac{2x^2-5}{7x^2+x-3} = \boxed{\frac27}$

because both numerator and denominator have the same degree (2), so their end behaviors are similar enough that the ratio of their coefficients determine the limit at infinity.

More precisely, we can divide through the expression uniformly by x ²,

$\displaystyle \lim_{x\to\infty} \frac{2x^2-5}{7x^2+x-3} = \lim_{x\to\infty} \frac{2-\dfrac5{x^2}}{7+\dfrac1x-\dfrac3{x^2}}$

Then each remaining rational term converges to 0 as x gets arbitrarily large, leaving 2 in the numerator and 7 in the denominator.

B) By the same reasoning,

$\displaystyle \lim_{x\to\infty} \frac{5x-3}{2x+1} = \boxed{\frac52}$

C) This time, the degree of the denominator exceeds the degree of the numerator, so it grows faster than x - 1. Dividing a number by a larger number makes for a smaller number. This means the limit will be 0:

$\displaystyle \lim_{x\to-\infty} \frac{x-1}{x^2+8} = \boxed{0}$

More precisely,

$\displaystyle \lim_{x\to-\infty} \frac{x-1}{x^2+8} = \lim_{x\to-\infty}\frac{\dfrac1x-\dfrac1{x^2}}{1+\dfrac8{x^2}} = \dfrac01 = 0$

D) Looks like this limit should read

$\displaystyle \lim_{t\to\infty}\frac{\sqrt{t}+t^2}{3t-t^2}$

which is just another case of (A) and (B); the limit would be

$\displaystyle \lim_{t\to\infty}\frac{\sqrt{t}+t^2}{3t-t^2} = -1$

That is,

$\displaystyle \lim_{t\to\infty}\frac{\sqrt{t}+t^2}{3t-t^2} = \lim_{t\to\infty}\frac{\dfrac1{t^{3/2}}+1}{\dfrac3t-1} = \dfrac1{-1} = -1$

However, in case you meant something else, such as

$\displaystyle \lim_{t\to\infty}\frac{\sqrt{t+t^2}}{3t-t^2}$

then the limit would be different:

$\displaystyle \lim_{t\to\infty}\frac{\sqrt{t^2}\sqrt{\dfrac1t+1}}{3t-t^2} = \lim_{t\to\infty}\frac{t\sqrt{\dfrac1t+1}}{3t-t^2} = \lim_{t\to\infty}\frac{\sqrt{\dfrac1t+1}}{3-t} = 0$

since the degree of the denominator is larger.

One important detail glossed over here is that

$\sqrt{t^2} = |t|$

for all real t. But since t is approaching *positive* infinity, we have t > 0, for which |t | = t.

E) Similar to (D) - bear in mind this has the same ambiguity I mentioned above, but in this case the limit's value is unaffected -

$\displaystyle \lim_{x\to\infty} \frac{x^4}{\sqrt{x^8+9}} = \lim_{x\to\infty}\frac{x^4}{\sqrt{x^8}\sqrt{1+\dfrac9{x^8}}} = \lim_{x\to\infty}\frac{x^4}{x^4\sqrt{1+\dfrac9{x^8}}} = \lim_{x\to\infty}\frac1{\sqrt{1+\dfrac9{x^8}}} = \boxed{1}$

Again,

$\sqrt{x^8} = |x^4|$

but x ⁴ is non-negative for real x.

F) Also somewhat ambiguous:

$\displaystyle \lim_{x\to\infty}\frac{\sqrt{x+5x^2}}{3x-1} = \lim_{x\to\infty}\frac{\sqrt{x^2}\sqrt{\dfrac1x+5}}{3x-1} = \lim_{x\to\infty}\frac{x\sqrt{\dfrac1x+5}}{3x-1} = \lim_{x\to\infty}\frac{\sqrt{\dfrac1x+5}}{3-\dfrac1x} = \dfrac{\sqrt5}3$

$\displaystyle \lim_{x\to\infty}\frac{\sqrt{x}+5x^2}{3x-1} = \lim_{x\to\infty}x \cdot \lim_{x\to\infty}\frac{\dfrac1{\sqrt x}+5x}{3x-1} = \lim_{x\to\infty}x \cdot \lim_{x\to\infty}\frac{\dfrac1{x^{3/2}}+5}{3-\dfrac1x} = \frac53\lim_{x\to\infty}x = \infty$

G) For a regular polynomial (unless you left out a denominator), the leading term determines the end behavior. In other words, for large x, x ⁴ is much larger than x ², so effectively

$\displaystyle \lim_{x\to\infty}(x^4-2x) = \lim_{x\to\infty}x^4 = \boxed{\infty}$

6 0

3 years ago