How do you find the Limit of #ln(x) / sqrtx # as x approaches infinity?

Answer 1

#ln(x)# grows more slowly than any positive power of #x# as #x->oo#.

Hence the limit is #0#

Quick answer

Note that the exponential function #e^x# grows faster than any polynomial, in particular #x^2#
As a result the inverse function #ln x# grows more slowly than the inverse function #sqrt(x)#
Here's #e^x# and #x^2# ...

graph{(y-e^x)(y-x^2) = 0 [-39.96, 40.04, -164.4, 235.6]}

Here's #ln x# and #x^(1/2)# ...

graph{(y - ln x)(y - x^(1/2)) = 0 [-96.4, 100.36, -4.56, 5.44]}

Hence:

#lim_(x->oo) (ln x)/(sqrt(x)) = 0#
Sign up to view the whole answer

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

Sign up with email
Answer 2

To find the limit of ln(x) / sqrt(x) as x approaches infinity, we can use L'Hôpital's rule. By applying this rule, we differentiate the numerator and denominator separately and then take the limit as x approaches infinity again.

Differentiating ln(x) with respect to x gives 1/x, and differentiating sqrt(x) with respect to x gives 1/(2*sqrt(x)).

Taking the limit as x approaches infinity of the differentiated numerator and denominator, we get:

lim(x→∞) [1/x] / [1/(2*sqrt(x))]

Simplifying this expression, we can multiply the numerator and denominator by 2*sqrt(x):

lim(x→∞) [2*sqrt(x)] / x

Now, we can cancel out the x in the denominator:

lim(x→∞) 2 / sqrt(x)

As x approaches infinity, the square root of x also approaches infinity. Therefore, the limit becomes:

lim(x→∞) 2 / ∞ = 0

Sign up to view the whole answer

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

Sign up with email
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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7