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This limit can be evaluated using either L'Hopital's Rule or a standard trigonometric limit.Method 1: Using L'Hopital's RuleAs x→0, tan(4x)→tan(0)=0 and 3x→0. Since the limit is of the indeterminate form 0/0, we can apply L'Hopital's Rule.L'Hopital's Rule states that lim (x→c) f(x)/g(x) = lim (x→c) f'(x)/g'(x), provided the latter limit exists.Derivative of the numerator f(x) = tan(4x): f'(x) = sec²(4x) * 4Derivative of the denominator g(x) = 3x: g'(x) = 3So, lim (x→0) (tan4x)/(3x) = lim (x→0) [4sec²(4x)] / 3Substitute x=0: [4sec²(40)] / 3 = [4sec²(0)] / 3 = (41)/3 = 4/3Method 2: Using the standard limit lim (u→0) (tan u)/u = 1We can rewrite the expression to use this standard limit:lim (x→0) (tan4x)/(3x) = lim (x→0) [ (tan4x) / (4x) ] * (4x) / (3x)= lim (x→0) [ (tan4x) / (4x) ] * (4/3)As x→0, 4x also approaches 0. So, lim (x→0) (tan4x)/(4x) = 1.Therefore, the limit is 1 * (4/3) = 4/3.