How do you find the limit #lim (pi^x-pi)/(pi^(2x)-pi^2)# as #x->1#?

Answer 1

#1/(2pi)#

#lim_(x->1) (pi^x-pi)/(pi^(2x)-pi^2)=lim_(x->1) (pi^x-pi)/((pi^(x)-pi)(pi^(x)+pi))=1/(2pi)#
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 lim (pi^x-pi)/(pi^(2x)-pi^2) as x approaches 1, we can use algebraic manipulation and the properties of limits. First, we can factor out pi from both the numerator and denominator:

(pi^x - pi) / (pi^(2x) - pi^2) = pi * (pi^(x-1) - 1) / (pi^2 * (pi^(2x-2) - 1))

Next, we can simplify further by canceling out the common factors of pi:

(pi^(x-1) - 1) / (pi^(2x-2) - 1)

Now, we can substitute x = 1 into the expression:

(pi^(1-1) - 1) / (pi^(2(1)-2) - 1) = (1 - 1) / (pi^0 - 1) = 0 / 0

At this point, we have an indeterminate form of 0/0. To evaluate this limit, we can apply L'Hôpital's rule by taking the derivative of the numerator and denominator with respect to x:

d/dx (pi^(x-1) - 1) / d/dx (pi^(2x-2) - 1)

Using the chain rule, we get:

(pi^(x-1) * ln(pi)) / (2 * pi^(2x-2) * ln(pi))

Now, we can substitute x = 1 into the derivative expression:

(pi^(1-1) * ln(pi)) / (2 * pi^(2(1)-2) * ln(pi)) = (1 * ln(pi)) / (2 * pi^0 * ln(pi)) = ln(pi) / (2 * 1 * ln(pi)) = 1/2

Therefore, the limit of (pi^x-pi)/(pi^(2x)-pi^2) as x approaches 1 is 1/2.

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