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What is the limit of # sqrt[x^2-2x+3 - sqrtx]# as x goes to infinity?

Answer 1

Charles Anctil

#lim_(xrarroo) sqrt[x^2-2x+3 - sqrtx] = oo#

# sqrt[x^2-2x+3 - sqrtx] = sqrt(x^2)sqrt(1-2/x+3/x^2=sqrtx/x^2)#

# = absx sqrt(1-2/x+3/x^2=sqrtx/x^2)#

As #x rarr oo#, the radicand goes to #1#, while #absx = x rarr oo#

Indeed, #y=x# is an oblique asymptote on the right for the graph of #f(x) = sqrt[x^2-2x+3 - sqrtx]#.

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Answer 2

Emily Ammerman

The limit of sqrt[x^2-2x+3 - sqrtx] as x goes to 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.