How do you evaluate the limit #sqrt(x^2+1)-2x# as x approaches #oo#?

Answer 1

See below.

Change the way the expression is written.

If we try to find the limit as written, we get the indeterminate form #oo-oo#.
We could change this to a fraction using #(sqrt(x^2+1)+2x)/(sqrt(x^2+1)+2x)#, but perhaps the following is simpler.
For #x !`= 0#, we have
#sqrtx^2+1) - 2x = sqrt(x^2(1+1/x^2))-2x#
# = sqrt(x^2)sqrt(1+1/x^2)-2x#
For positive #x#, #sqrt(x^2) = x# so
#lim_(xrarroo) (sqrt(x^2+1) - 2x) = lim_(xrarroo)(xsqrt(1+1/x^2)-2x)#
# = lim_(xrarroo)(x(sqrt(1+1/x^2) - 2))#
Which has the form #oo * (1-2)#

Therefore,

#lim_(xrarroo) (sqrt(x^2+1) - 2x) = -oo#
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Answer 2

To evaluate the limit of sqrt(x^2+1)-2x as x approaches infinity, we can simplify the expression. By multiplying the numerator and denominator by the conjugate, sqrt(x^2+1)+2x, we can eliminate the square root in the numerator. This simplifies the expression to (sqrt(x^2+1)-2x)(sqrt(x^2+1)+2x) / (sqrt(x^2+1)+2x).

Expanding the numerator gives (x^2+1) - (4x^2) = -3x^2 + 1.

As x approaches infinity, the -3x^2 term dominates, and the limit becomes -3x^2 / (sqrt(x^2+1)+2x).

Since the degree of the numerator is greater than the degree of the denominator, the limit as x approaches infinity is negative infinity.

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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.

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.

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.

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.

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