What is the limit as x approaches infinity of (ln(7e^{3x}+2x^5))/(4x+3) ?

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limit as x approaches infinity of (ln(7e^{3x}+2x^5))/(4x+3)

x \to \infty lim (\frac{ln ( 7 e ^{3 x} + 2 x ^{5} )}{4 x + 3})

\frac{3}{4}

Decimal

0.75

Solution steps

x \to \infty lim (\frac{ln ( 7 e ^{3 x} + 2 x ^{5} )}{4 x + 3})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{\frac{21 e ^{3 x} + 10 x ^{4}}{7 e ^{3 x} + 2 x ^{5}}}{4})

= x \to \infty lim (\frac{\frac{21 e ^{3 x} + 10 x ^{4}}{7 e ^{3 x} + 2 x ^{5}}}{4})

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= x \to \infty lim (\frac{21 e ^{3 x} + 10 x ^{4}}{4 ( 7 e ^{3 x} + 2 x ^{5} )})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{63 e ^{3 x} + 40 x ^{3}}{4 ( 21 e ^{3 x} + 10 x ^{4} )})

= x \to \infty lim (\frac{63 e ^{3 x} + 40 x ^{3}}{4 ( 21 e ^{3 x} + 10 x ^{4} )})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{189 e ^{3 x} + 120 x ^{2}}{4 ( 63 e ^{3 x} + 40 x ^{3} )})

= x \to \infty lim (\frac{189 e ^{3 x} + 120 x ^{2}}{4 ( 63 e ^{3 x} + 40 x ^{3} )})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{567 e ^{3 x} + 240 x}{4 ( 189 e ^{3 x} + 120 x ^{2} )})

= x \to \infty lim (\frac{567 e ^{3 x} + 240 x}{4 ( 189 e ^{3 x} + 120 x ^{2} )})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{1701 e ^{3 x} + 240}{4 ( 567 e ^{3 x} + 240 x )})

= x \to \infty lim (\frac{1701 e ^{3 x} + 240}{4 ( 567 e ^{3 x} + 240 x )})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{5103 e ^{3 x}}{4 ( 1701 e ^{3 x} + 240 )})

= x \to \infty lim (\frac{5103 e ^{3 x}}{4 ( 1701 e ^{3 x} + 240 )})

Apply L'Hopital's Rule

: x \to \infty lim (\frac{15309 e ^{3 x}}{20412 e ^{3 x}})

= x \to \infty lim (\frac{15309 e ^{3 x}}{20412 e ^{3 x}})

\frac{15309 e ^{3 x}}{20412 e ^{3 x}} = \frac{3}{4}

= x \to \infty lim (\frac{3}{4})

x \to a lim c = c

= \frac{3}{4}

x \to - 1 + lim (x^{3})

x \to - \infty lim (\frac{1}{1 x})

x \to 0 lim (\frac{1 - cos ^{2} ( x - 2 )}{3 x ^{2} - 12 x + 12})

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

x \to \infty lim ((\frac{4 x + 3}{x ^{2} + 3 x - 4}) sin (3 x - 2 π))

What is the limit as x approaches infinity of (ln(7e^{3x}+2x^5))/(4x+3) ?
The limit as x approaches infinity of (ln(7e^{3x}+2x^5))/(4x+3) is 3/4