What is the limit as x approaches infinity of (x+7)/(7+7^{x+1)(x+6)} ?

The limit as x approaches infinity of (x+7)/(7+7^{x+1)(x+6)} is 0

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limit as x approaches infinity of (x+7)/(7+7^{x+1)(x+6)}

x \to \infty lim (\frac{x + 7}{7 + 7 ^{x + 1} ( x + 6 )})

0

Solution steps

x \to \infty lim (\frac{x + 7}{7 + 7 ^{x + 1} ( x + 6 )})

Divide by

7^{x + 1} : \frac{\frac{x}{7 ^{x + 1}} + \frac{1}{7 ^{x}}}{\frac{1}{7 ^{x}} + x + 6}

= x \to \infty lim (\frac{\frac{x}{7 ^{x + 1}} + \frac{1}{7 ^{x}}}{\frac{1}{7 ^{x}} + x + 6})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

With the exception of indeterminate form

= \frac{lim _{x \to \infty} ( \frac{x}{7 ^{x + 1}} + \frac{1}{7 ^{x}} )}{lim _{x \to \infty} ( \frac{1}{7 ^{x}} + x + 6 )}

x \to \infty lim (\frac{x}{7 ^{x + 1}} + \frac{1}{7 ^{x}}) = 0

x \to \infty lim (\frac{1}{7 ^{x}} + x + 6) = \infty

= \frac{0}{\infty}

Apply Infinity Property:

\frac{0}{\infty} = 0

= 0

x \to - 3 lim (\frac{2 x - 1}{x - 3})

x \to 2 lim (3 x^{2} - 6 x + 8)

x \to a lim (\frac{3 a x - 2 a + 2 x - 3 a ^{2}}{5 a x + a - 5 a ^{2} - x})

x \to 0 lim (sin (π))

t \to 0 lim (\frac{sin ( - 3 t )}{t})

What is the limit as x approaches infinity of (x+7)/(7+7^{x+1)(x+6)} ?
The limit as x approaches infinity of (x+7)/(7+7^{x+1)(x+6)} is 0