What is the limit as x approaches infinity of (x^2+2^x)/((x+1)^2+2^{x+1)} ?

The limit as x approaches infinity of (x^2+2^x)/((x+1)^2+2^{x+1)} is 1/2

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limit as x approaches infinity of (x^2+2^x)/((x+1)^2+2^{x+1)}

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

\frac{1}{2}

Decimal Notation

0.5

Solution steps

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

Apply L'Hopital's Rule

: x \to \infty lim (\frac{2 x + 2 ^{x} ln ( 2 )}{2 ( x + 1 ) + ln ( 2 ) \cdot 2 ^{x + 1}})

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

Apply L'Hopital's Rule

: x \to \infty lim (\frac{2 + ln ^{2} ( 2 ) \cdot 2 ^{x}}{2 + ln ^{2} ( 2 ) \cdot 2 ^{x + 1}})

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

Apply L'Hopital's Rule

: x \to \infty lim (\frac{ln ^{3} ( 2 ) \cdot 2 ^{x}}{ln ^{3} ( 2 ) \cdot 2 ^{x + 1}})

= x \to \infty lim (\frac{ln ^{3} ( 2 ) \cdot 2 ^{x}}{ln ^{3} ( 2 ) \cdot 2 ^{x + 1}})

\frac{ln ^{3} ( 2 ) \cdot 2 ^{x}}{ln ^{3} ( 2 ) \cdot 2 ^{x + 1}} = \frac{1}{2}

= x \to \infty lim (\frac{1}{2})

x \to a lim c = c

= \frac{1}{2}

\frac{d}{d t} (\frac{3 e ^{3 α t} xα}{4})

\int (x + 5) 7 d x

\int \frac{3}{1 - y} d y

y = - x^{2} + 1, (- 1, 1)

\frac{d}{d x} (x^{3} y^{2})

What is the limit as x approaches infinity of (x^2+2^x)/((x+1)^2+2^{x+1)} ?
The limit as x approaches infinity of (x^2+2^x)/((x+1)^2+2^{x+1)} is 1/2