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How do you find the limit of # (x^7 + 8x^3)/(4x^8 + 6x^12)# as x approaches 0?

Answer 1

Scarlett Allison

#lim_(x->0)(x^7+8x^3)/(4x^8+6x^12)#

#lim_(x->0) (x^3(x^4 +8))/(x^3(4x^5+6x^9)#

#lim_(x->0) (x^4 +8)/(4x^5+6x^9)#

#=8/0 = oo#

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Answer 2

Scarlett Allison

The limit does not exist.

Let #f(x) = (x^7+8x^3)/(6x^12+4x^8) = (x^4+8)/(6x^9+4x^5)#

#lim_(xrarr0^-)f(x) = -oo#

#lim_(xrarr0^+)f(x) = oo#

Therefore,

#lim_(xrarr0)f(x)# #" "# Does Not Exist.

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Answer 3

Aurora Alvarado

To find the limit of (x^7 + 8x^3)/(4x^8 + 6x^12) as x approaches 0, we can simplify the expression by factoring out the highest power of x in both the numerator and denominator.

Factoring out x^3 from the numerator, we get x^3(x^4 + 8).

Factoring out x^8 from the denominator, we get x^8(4 + 6x^4).

Canceling out the common factors, we have (x^3(x^4 + 8))/(x^8(4 + 6x^4)).

Now, as x approaches 0, all the terms with x in the expression tend to 0.

Therefore, the limit of (x^7 + 8x^3)/(4x^8 + 6x^12) as x approaches 0 is 0.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.