evaluate the following limit. use lhôpitals rule when it is convenient and applicable.\n\n$$\\lim_{x \\to 0}…

evaluate the following limit. use lhôpitals rule when it is convenient and applicable.\n\n$$\\lim_{x \\to 0} \\frac{4 \\sin 3x}{7x}$$\n\nuse lhôpitals rule to rewrite the given limit so that it is not an indeterminate form.\n\n$$\\lim_{x \\to 0} \\frac{4 \\sin 3x}{7x}=\\lim_{x \\to 0} \\left(\\square\\right)$$\n\nevaluate the limit.\n\n$$\\lim_{x \\to 0} \\frac{4 \\sin 3x}{7x}=\\square$$ (type an exact answer.)

evaluate the following limit. use lhôpitals rule when it is convenient and applicable.\n\n$$\\lim_{x \\to 0} \\frac{4 \\sin 3x}{7x}$$\n\nuse lhôpitals rule to rewrite the given limit so that it is not an indeterminate form.\n\n$$\\lim_{x \\to 0} \\frac{4 \\sin 3x}{7x}=\\lim_{x \\to 0} \\left(\\square\\right)$$\n\nevaluate the limit.\n\n$$\\lim_{x \\to 0} \\frac{4 \\sin 3x}{7x}=\\square$$ (type an exact answer.)

Answer

Explanation:

Step1: Apply L'Hôpital's Rule

L'Hôpital's Rule states that if (\lim_{x\rightarrow a}\frac{f(x)}{g(x)}) is in the indeterminate form (\frac{0}{0}) or (\frac{\pm\infty}{\pm\infty}), then (\lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\lim_{x\rightarrow a}\frac{f^{\prime}(x)}{g^{\prime}(x)}). For (f(x) = 4\sin(3x)) and (g(x)=7x), (f^{\prime}(x)=4\times3\cos(3x)=12\cos(3x)) and (g^{\prime}(x) = 7). So, (\lim_{x\rightarrow0}\frac{4\sin(3x)}{7x}=\lim_{x\rightarrow0}\frac{12\cos(3x)}{7}).

Step2: Evaluate the limit

Substitute (x = 0) into (\frac{12\cos(3x)}{7}). Since (\cos(0)=1), we have (\frac{12\cos(0)}{7}=\frac{12\times1}{7}).

Answer:

(\lim_{x\rightarrow0}\frac{4\sin(3x)}{7x}=\frac{12}{7})