Anonymous
Anonymous asked in Science & MathematicsMathematics · 2 months ago

# Find the limit, if it exists. (If an answer does not exist, enter DNE.) lim x → ∞ x5/ √(x10 + 7)?

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• ?
Lv 7
2 months ago

limit [(x^5)/sqr(x^10+7)]=? If so, then

x->inf.

the LHS=

limit [(x^5)/{(x^5)sqr[1+7/(x^10)]}=1.

x->inf.

• 2 months ago

lim ........x^5

x->∞ ------------

........√(x^10 + 7)

...........x^5/x^5

=lim ------------

x->∞ √(x^10/x^10 + 7/x^10)

.............1

=lim -------------

x->∞...√(1)

= 1/1

• ?
Lv 6
2 months ago

[lim x→inf] x^5 / √(x^10 + 7) ; we know x is positive

= [lim x→inf] [x^5/x^5] / [√(x^10 + 7)/x^5]

= [lim x→inf] 1 / [√((x^10 + 7)/x^10)]

= [lim x→inf] 1 / [√(1 + 7/x^10)]

= 1 ; because 7/x^10 approaches 0 when x goes to inf.

• ?
Lv 7
2 months ago

Factor out x5 from top & bottom to get:

(x5/x5) * 1 / √( 1 + 7/x10) =

1 / √( 1 + 7/x10).

This is much simpler, no?

As x --> ∞ this gives?

Done!

• 2 months ago

You can use the "sandwich method"

a) remember that on the way to a limit, as x->∞, you use large positive integers, for x

b) if you make the denominator larger, then the whole fraction is smaller (and vice-versa: a smaller denominator means a larger fraction.

---

x^5 / √(x^10 + 7)

The denominator is √(x^10 + 7)

x^10 < x^10 + 7

Therefore, the fraction x^5 / √x^10

is larger than the original fraction (it has a smaller denominator)

Lim x->∞ { x^5 / √x^10 } =

Lim x->∞ { x^5 / x^5 } = 1

The limit of the original fraction cannot be larger than 1.

---

Since x is positive, adding x values in the denominator can only make it larger;

larger denominator = smaller fraction

√(x^10 + 8x^5 + 16) > √(x^10 + 7)

√[(x^5 + 4)^2]  > √(x^10 + 7)

(x^5 + 4) > √(x^10 + 7)

Therefore, the fraction x^5 / (x^5 + 4) is smaller than the original fraction

The limit of that fraction (as x goes to infinity) is 1

The limit of the original fraction cannot be smaller than 1.

----

Cannot be bigger that 1

Cannot be smaller than 1

Then it must be 1

• 2 months ago

The customary way to write an exponent using a typical keyboard is with the caret " ^ " -- like, x^2 for "x squared" and x^3 for "x cubed."

So I'll assume you meant

lim (x --> infinity) [x^5 / sqrt(x^10 + 7)].

The limit is 1, because when x is very large, the square root of x^10 becomes nearly indistinguishable from x^5.