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How do you find the limit of #(x+x^3-x^5)/(1-x^2-x^4)# as x approaches infinity?

Answer 1

Christopher Agresta

#oo#

#L = lim_(x to oo) (x+x^3-x^5)/(1-x^2-x^4)#

#=lim_(x to oo) x(1+x^2-x^4)/(1-x^2-x^4)#

#=lim_(x to oo) x(1/x^4+1/x^2-1)/(1/x^4-1/x^2-1)#

..... and as #= lim_(x to oo) 1/x^n = 0, n > 0#

this has a slant asymptote #y = x#

#therefore L = lim_{x to oo} x = oo#

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Answer 2

Samantha Adkison

To find the limit of (x+x^3-x^5)/(1-x^2-x^4) as x approaches infinity, we can divide both the numerator and denominator by x^5. This gives us (1/x^4 + 1/x^2 - 1)/(1/x^5 - 1/x^3 - 1).

As x approaches infinity, all terms with x in the denominator tend to zero. Therefore, the limit simplifies to (0 + 0 - 1)/(0 - 0 - 1), which is equal to -1.

Hence, the limit of (x+x^3-x^5)/(1-x^2-x^4) as x approaches infinity is -1.

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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.