What is the limit as x approaches infinity of (5x)/(5x+3x^{1/x)} ?

The limit as x approaches infinity of (5x)/(5x+3x^{1/x)} is 1

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limit as x approaches infinity of (5x)/(5x+3x^{1/x)}

x \to \infty lim (\frac{5 x}{5 x + 3 x ^{\frac{1}{x}}})

1

Solution steps

x \to \infty lim (\frac{5 x}{5 x + 3 x ^{\frac{1}{x}}})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= 5 \cdot x \to \infty lim (\frac{x}{5 x + 3 x ^{\frac{1}{x}}})

Factor out

5 x

from

5 x + 3 x^{\frac{1}{x}} : 5 x (1 + \frac{3 x ^{\frac{1}{x}}}{5 x})

= 5 \cdot x \to \infty lim \frac{x}{5 x ( 1 + \frac{3 x ^{\frac{1}{x}}}{5 x} )}

Simplify

\frac{x}{5 x ( 1 + \frac{3 x ^{\frac{1}{x}}}{5 x} )} : \frac{1}{3 x ^{\frac{1 - x}{x}} + 5}

= 5 \cdot x \to \infty lim (\frac{1}{3 x ^{\frac{1 - x}{x}} + 5})

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

= 5 \cdot \frac{lim _{x \to \infty} ( 1 )}{lim _{x \to \infty} ( 3 x ^{\frac{1 - x}{x}} + 5 )}

x \to \infty lim (1) = 1

x \to \infty lim (3 x^{\frac{1 - x}{x}} + 5) = 5

= 5 \cdot \frac{1}{5}

Simplify

5 \cdot \frac{1}{5} : 1

= 1

f (x) = 2 x - 3, 4, 0

f (x) = \frac{2}{x}, at x = 5

e^{3 x}

\int \frac{x + 5}{x + 6} d x

\frac{d}{d x} (cos (x) - sin (x))

What is the limit as x approaches infinity of (5x)/(5x+3x^{1/x)} ?
The limit as x approaches infinity of (5x)/(5x+3x^{1/x)} is 1