Can someone help find limit using l'hopital rule
lim x → ∞ ((3x-4)/(3x+2))^(3x+1), I'm trying to solve by adding ln function to both side.
To find the limit of the given expression using L'Hôpital's rule, we can differentiate both the numerator and denominator repeatedly until we reach a manageable form. Let's go step by step:
Step 1: Take the natural logarithm (ln) of both sides of the expression to simplify the exponent:
ln[lim(x → ∞) ((3x - 4)/(3x + 2))^(3x + 1)]
Step 2: Simplify the exponent using the logarithmic property:
(3x + 1) ln[(3x - 4)/(3x + 2)]
Step 3: Apply L'Hôpital's rule by differentiating the numerator and denominator with respect to x.
[(d/dx) (3x - 4)] / [(d/dx) (3x + 2)]
Step 4: Evaluate the derivatives:
[3/(3x + 2)] / [3/(3)]
Step 5: Simplify the expression:
[1/(x + 2/3)] / 1
Step 6: Take the limit as x approaches infinity:
[1/(∞ + 2/3)] / 1 = [1/(∞)] / 1 = 0
Therefore, the limit of the given expression using L'Hôpital's rule is 0.