Evaluate Using L'Hospital's Rule

Problem

(lim_x→8)(ln(√(,x))/(x2))

Solution

1. Identify the limit expression and check for indeterminate forms.

2. Substitute the value x=8 into the expression to see if L'Hospital's Rule is necessary.

ln(√(,8))/8

3. Simplify the numerator using the properties of logarithms, specifically ln(xa)=a*ln(x)

ln(√(,x))=ln(x(1/2))=1/2*ln(x)

4. Evaluate the limit directly since the expression is continuous at x=8 and does not result in an indeterminate form like 0/0 or ∞/∞

(lim_x→8)((1/2*ln(x))/(x2))

5. Calculate the final value by plugging in x=8

(1/2*ln(8))/8=ln(8)/(2⋅64)=ln(8)/128

6. Simplify the logarithm further if desired, noting that 8=2 so ln(8)=3*ln(2)

(3*ln(2))/128

Note: L'Hospital's Rule is not applicable here because the limit does not result in an indeterminate form.

Final Answer

(lim_x→8)(ln(√(,x))/(x2))=(3*ln(2))/128

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