lim_(x->0^+)(tan(9x))^x
To find the limit of the function as x approaches 0 from the right-hand side, we can apply some algebraic manipulation and use the laws of limits. Here's the step-by-step process:
Step 1: Rewrite the expression in a more manageable form.
The function can be rewritten as (tan(9x))^(1/x) since raising a number to the power of 1/x is equivalent to taking the x-th root of that number.
Step 2: Take the natural logarithm of both sides to simplify the expression.
ln((tan(9x))^(1/x)) = ln(tan(9x))/x
Step 3: Apply the logarithmic property to break down the expression.
ln(tan(9x))/x = (ln(tan(9x)) - ln(1))/x
Step 4: Apply the quotient rule for logarithms.
(ln(tan(9x)) - ln(1))/x = (ln(tan(9x)/1))/x = ln(tan(9x))/x
Step 5: Take the limit as x approaches 0.
lim(x->0^+) ln(tan(9x))/x
Now, let's evaluate this limit step-by-step:
Step 6: Apply the properties of logarithms.
lim(x->0^+) ln(tan(9x))/x = ln(lim(x->0^+) tan(9x))/x
Step 7: Apply the limit of the inside function.
lim(x->0^+) tan(9x) = tan(0) = 0 (the limit of tangent as x approaches 0 is 0)
Step 8: Substitute the obtained limit back into the expression.
ln(lim(x->0^+) tan(9x))/x = ln(0)/x
Step 9: Apply the limit of ln(0)/x as x approaches 0.
As x approaches 0, ln(0)/x has the form 0/0, which is an indeterminate form.
To apply L'Hôpital's rule, take the derivative of the numerator and denominator:
Step 10: Taking the derivative of ln(0) and x.
d/dx (ln(0)) = 0, and d/dx (x) = 1.
Step 11: Apply L'Hôpital's rule.
lim(x->0^+) ln(0)/x = lim(x->0^+) 0/1 = 0
Therefore, the limit of (tan(9x))^x as x approaches 0 from the right-hand side is 0.