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

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

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.

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

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.

- How do you evaluate the limit #(x^4+3^x)/(x^5+1)# as x approaches #oo#?
- How do you use the Squeeze Theorem to find #limsin(1/x)# as x approaches zero?
- How do you evaluate the limit #(2p+4)/(3p)# as p approaches #-2#?
- For what values of x, if any, does #f(x) = 1/x-tan(x) # have vertical asymptotes?
- What is the limit of #(e^x + x)^(1/x)# as x approaches infinity?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7