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

Answer 1

Isaiah Allen

#lim_(x->-oo) (5x^2-2)^(1/2)/(x+3) = -sqrt(5)#

Note that if #x < 0# then #sqrt(x^2) = -x#

So:

#lim_(x->-oo) (5x^2-2)^(1/2)/(x+3) = lim_(x->-oo)(-x(5-2/x^2)^(1/2))/(x+3)#

#color(white)(lim_(x->-oo) (5x^2-2)^(1/2)/(x+3)) = lim_(x->-oo)(-(5-2/x^2)^(1/2))/(1+3/x)#

#color(white)(lim_(x->-oo) (5x^2-2)^(1/2)/(x+3)) = (-sqrt(5))/1 = -sqrt(5)#

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Answer 2

Axel Alvarez

To find the limit of ((5x^2-2)^(1/2))/(x+3) as x approaches -oo, we can use the concept of limits. By dividing both the numerator and denominator by x, we can rewrite the expression as (5 - 2/x^2)^(1/2)/(1 + 3/x). As x approaches -oo, both 2/x^2 and 3/x approach 0. Therefore, the expression simplifies to (5)^(1/2)/(1). Simplifying further, we get the limit as x approaches -oo to be √5.

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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.