What is the limit as x approaches infinity of 10x[ln(x+9)-ln(x)] ?

The limit as x approaches infinity of 10x[ln(x+9)-ln(x)] is 90

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

limit as x approaches infinity of 10x[ln(x+9)-ln(x)]

x \to \infty lim (10 x [ln (x + 9) - ln (x)])

90

Solution steps

x \to \infty lim (10 x [ln (x + 9) - ln (x)])

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

= 10 \cdot x \to \infty lim (x [ln (x + 9) - ln (x)])

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (x + 9) - ln (x) = ln (\frac{x + 9}{x})

= 10 \cdot x \to \infty lim (x [ln (\frac{x + 9}{x})])

Simplify

= 10 \cdot x \to \infty lim (x ln (\frac{x + 9}{x}))

Rewrite for L'Hopital

= 10 \cdot x \to \infty lim (\frac{ln ( \frac{9 + x}{x} )}{\frac{1}{x}})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{- \frac{9}{x ( x + 9 )}}{- \frac{1}{x ^{2}}})

= 10 \cdot x \to \infty lim (\frac{- \frac{9}{x ( x + 9 )}}{- \frac{1}{x ^{2}}})

Simplify

\frac{- \frac{9}{x ( x + 9 )}}{- \frac{1}{x ^{2}}} : \frac{9 x}{x + 9}

= 10 \cdot x \to \infty lim (\frac{9 x}{x + 9})

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

= 10 \cdot 9 \cdot x \to \infty lim (\frac{x}{x + 9})

Divide by highest denominator power:

\frac{1}{1 + \frac{9}{x}}

= 10 \cdot 9 \cdot x \to \infty lim (\frac{1}{1 + \frac{9}{x}})

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

= 10 \cdot 9 \cdot \frac{lim _{x \to \infty} ( 1 )}{lim _{x \to \infty} ( 1 + \frac{9}{x} )}

x \to \infty lim (1) = 1

x \to \infty lim (1 + \frac{9}{x}) = 1

= 10 \cdot 9 \cdot \frac{1}{1}

Simplify

= 90

x \to 5 lim (x^{2} - 12 x)

x \to 6 lim (\frac{6 - ∣ x ∣}{6 + x})

y \to 0 lim ((cos (8 y)) \frac{1}{y ^{2}})

x \to 0 lim (x \cdot co tan (x))

x \to 0 lim (x (5^{\frac{1}{x}} - 1))

What is the limit as x approaches infinity of 10x[ln(x+9)-ln(x)] ?
The limit as x approaches infinity of 10x[ln(x+9)-ln(x)] is 90