please give me algorithm how to solve the
lim sqrt(tan(0.25ïx))^(1/(x^2-1))
x->1
To solve the limit of the given expression, we can start by simplifying the expression inside the square root:
1. Simplify the expression inside the square root:
a. Use the identity tan(0.5x) = sin(x) / cos(x).
b. tan(0.25x) = tan(0.5 * 0.5x) = sin(0.5x) / cos(0.5x).
c. So, sqrt(tan(0.25x)) = sqrt(sin(0.5x) / cos(0.5x)).
Now, let's evaluate the limit:
2. Rewrite the expression in terms of x:
lim (x -> 1) sqrt(tan(0.25x))^(1/(x^2-1)).
3. Use the fact that lim a^b = e^(b * ln(a)) to rewrite exponential as a logarithmic form:
lim (x -> 1) e^(ln(sqrt(tan(0.25x))))^(1/(x^2-1)).
4. Rewrite the expression in terms of natural logarithm:
lim (x -> 1) e^((ln(sqrt(tan(0.25x)))) * (1/(x^2-1))).
5. Expand the expression:
lim (x -> 1) e^((ln(sqrt(tan(0.25x))) - ln(e)) / (x^2-1)).
6. Simplify the expression:
lim (x -> 1) e^((ln(sqrt(tan(0.25x))) - 1) / (x^2-1)).
7. Apply the limit to the simplified expression:
lim (x -> 1) e^((ln(sqrt(tan(0.25x))) - 1) / (x-1) * (x+1)).
8. Simplify further (using L'Hôpital's rule):
a. Differentiate the numerator and denominator:
d/dx (ln(sqrt(tan(0.25x))) - 1) = (cos(0.25x) / (8 * tan(0.25x)^(3/2))) - 0
d/dx (x-1) * (x+1) = 1 * (x+1) + (x-1) * 1 = 2x.
b. Evaluate the new expression at x = 1:
(cos(0.25(1)) / (8 * tan(0.25(1))^(3/2))) / (2 * 1) = cos(0.25) / (16 * tan(0.25)^(3/2)).
Finally, to get the numerical value of the limit, we need to calculate the value of cos(0.25) and tan(0.25) and substitute them into the expression obtained in step 8.