guess the value of the limit\n lim_{x\rightarrowinfty}left(\frac{x^{6}}{6^{x}}\right) \nby evaluating the…

guess the value of the limit\n lim_{x\rightarrowinfty}left(\frac{x^{6}}{6^{x}}\right) \nby evaluating the function ( f(x)=\frac{x^{6}}{6^{x}} ) for ( x = 0,1,2,3,4,5,6,7,8,9,10,20,50 ), and ( 100 ). use a graph of ( f ) to support your guess.

guess the value of the limit\n lim_{x\rightarrowinfty}left(\frac{x^{6}}{6^{x}}\right) \nby evaluating the function ( f(x)=\frac{x^{6}}{6^{x}} ) for ( x = 0,1,2,3,4,5,6,7,8,9,10,20,50 ), and ( 100 ). use a graph of ( f ) to support your guess.

Answer

Explanation:

Step1: Evaluate the function at given points

When (x = 0), (f(0)=\frac{0^{6}}{6^{0}}=\frac{0}{1} = 0). When (x = 1), (f(1)=\frac{1^{6}}{6^{1}}=\frac{1}{6}\approx0.167). When (x = 2), (f(2)=\frac{2^{6}}{6^{2}}=\frac{64}{36}\approx1.778). When (x = 3), (f(3)=\frac{3^{6}}{6^{3}}=\frac{729}{216}\approx3.375). When (x = 4), (f(4)=\frac{4^{6}}{6^{4}}=\frac{4096}{1296}\approx3.161). When (x = 5), (f(5)=\frac{5^{6}}{6^{5}}=\frac{15625}{7776}\approx2.01). When (x = 6), (f(6)=\frac{6^{6}}{6^{6}} = 1). When (x = 7), (f(7)=\frac{7^{6}}{6^{7}}=\frac{117649}{282110.99}\approx0.417). When (x = 8), (f(8)=\frac{8^{6}}{6^{8}}=\frac{262144}{1679616}\approx0.156). When (x = 9), (f(9)=\frac{9^{6}}{6^{9}}=\frac{531441}{10077696}\approx0.053). When (x = 10), (f(10)=\frac{10^{6}}{6^{10}}=\frac{1000000}{60466176}\approx0.017). When (x = 20), (f(20)=\frac{20^{6}}{6^{20}}=\frac{64000000}{3656158440062976}\approx1.75\times 10^{-11}). When (x = 50), (f(50)=\frac{50^{6}}{6^{50}}=\text{very small positive - number}). When (x = 100), (f(100)=\frac{100^{6}}{6^{100}}=\text{extremely small positive - number}).

Step2: Analyze the trend

As (x) gets larger and larger, the value of (f(x)=\frac{x^{6}}{6^{x}}) approaches (0).

Answer:

(0)