How do you evaluate # [ ( ln x ) / (x^2 + x - 2 )]# as x approaches 1?

Answer 1

#lim_(x-> 1) (Ln(x))/(x^2+x-2) = 1/3#

If we evaluate the expression at the limit, we get #Ln(1) / (1^2 +1 -2 ) = 0/0#
This #0/0# result is a L'Hopital indeterminate form, it does not mean the limit does not exist, it means the limit does exist but needs to be derived using L'Hopitals Rule.

To find the limit using L'Hopitals rule simple evaluate the same limit for the derivative of the numerator over the derivative of the denominator.

So #lim_(x-> 1) ((Ln(x))')/(((x^2+x-2))') = (1/x)/(2x+1) = (1/1)/(2(1)+1) = 1/3#
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 evaluate the expression [(ln x) / (x^2 + x - 2)] as x approaches 1, we substitute 1 into the expression. This gives us [(ln 1) / (1^2 + 1 - 2)]. Since ln 1 is equal to 0, the expression simplifies to 0 / 0. This is an indeterminate form, so we need to apply L'Hôpital's rule. By taking the derivative of the numerator and denominator separately, we get [(1/x) / (2x + 1)]. Substituting x = 1 into this new expression gives us [(1/1) / (2(1) + 1)], which simplifies to 1/3. Therefore, as x approaches 1, the expression [(ln x) / (x^2 + x - 2)] evaluates to 1/3.

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