Home > Calculus > Determining Limits Algebraically

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

Answer 1

Isaiah Allen

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

When #a > 0#, #sqrt(a^2) = a#

So:

#lim_(x->oo) x/sqrt(x^2-x)#

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

#=lim_(x->oo) color(red)(cancel(color(black)(x)))/(color(red)(cancel(color(black)(x))) sqrt(1-1/x))#

#=lim_(x->oo) 1/sqrt(1-1/x)#

#= 1/sqrt(1 + 0) = 1#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Axel Alvarez

To find the limit of x / sqrt(x^2-x) as x approaches infinity, we can simplify the expression by multiplying both the numerator and denominator by 1/sqrt(x^2). This gives us the limit of (x / sqrt(x^2)) / sqrt(1 - 1/x) as x approaches infinity. Simplifying further, we have the limit of (x / |x|) / sqrt(1 - 1/x) as x approaches infinity. Since x approaches infinity, |x| is equal to x. Therefore, the expression simplifies to 1 / sqrt(1 - 1/x) as x approaches infinity. As x approaches infinity, 1/x approaches 0, so the expression becomes 1 / sqrt(1 - 0), which is equal to 1. Hence, the limit of x / sqrt(x^2-x) as x approaches infinity is 1.

By signing up, you agree to our Terms of Service and Privacy Policy

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.