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

Answer 1

Charles Anctil

#lim_{x to oo} sqrt(x^2 + x + 1) / x#

#= lim_{x to oo} sqrt(1 + 1/x + 1/x^2) / 1#

#=1#

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Answer 2

Isabella Ackerman

To find the limit of sqrt(x^2 + x + 1) / x as x approaches infinity, we can simplify the expression by dividing both the numerator and denominator by x. This gives us sqrt(1 + 1/x + 1/x^2). As x approaches infinity, both 1/x and 1/x^2 approach zero. Therefore, the limit simplifies to sqrt(1 + 0 + 0), which is equal to 1. Hence, the limit of sqrt(x^2 + x + 1) / x 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.