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How do you find the limit of #sqrt(4x^2-1) / x^2# as x approaches #oo#?

Answer 1

Charles Anctil

#0#

Combine all the terms into the square root:

#lim_(xrarroo)sqrt(4x^2-1)/x^2=lim_(xrarroo)sqrt(4x^2-1)/sqrt(x^4)#

#=lim_(xrarroo)sqrt((4x^2-1)/x^4)=lim_(xrarroo)sqrt(((4x^2-1)/x^4)/(x^4/x^4))#

#=lim_(xrarroo)sqrt((4/x^2-1/x^4)/1)=lim_(xrarroo)sqrt(4/x^2-1/x^4)#

We can now evaluate the limit.

#=sqrt(0-0)=0#

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Answer 2

William Abdalla

To find the limit of sqrt(4x^2-1) / x^2 as x approaches infinity, we can use the concept of limits.

First, we simplify the expression by dividing both the numerator and denominator by x^2:

sqrt(4x^2-1) / x^2 = (sqrt(4x^2-1)) / (x^2)

Next, as x approaches infinity, we can ignore the -1 term in the numerator since it becomes negligible compared to the large value of x^2.

Therefore, the expression simplifies to:

sqrt(4x^2) / x^2 = (2x) / x^2 = 2 / x

As x approaches infinity, the value of 2/x approaches 0.

Hence, the limit of sqrt(4x^2-1) / x^2 as x approaches infinity is 0.

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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.