What is the limit as x approaches infinity of (sqrt(x^2+2x)-x)^{6/x ln(cos(1/x))} ?

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limit as x approaches infinity of (sqrt(x^2+2x)-x)^{6/x ln(cos(1/x))}

x \to \infty lim ((x^{2} + 2 x - x)^{\frac{6}{x} l n (c o s (\frac{1}{x}))})

1

Solution steps

x \to \infty lim ((x^{2} + 2 x - x)^{\frac{6}{x} l n (c o s (\frac{1}{x}))})

Apply exponent rules

= x \to \infty lim (e^{\frac{6}{x} l n (c o s (\frac{1}{x})) l n (x^{2} + 2 x - x)})

Apply the Limit Chain Rule:

1

= 1

x \to \infty lim (\frac{x \cdot e ^{- x}}{x})

x \to 2 lim (\frac{3 x}{5 + 8 x})

x \to \frac{π}{4} lim (\frac{x}{tan ( x )})

x \to \infty lim (3^{2 x})

x \to 1 - lim (\frac{3 x + 65 - 4}{x + 1})

What is the limit as x approaches infinity of (sqrt(x^2+2x)-x)^{6/x ln(cos(1/x))} ?
The limit as x approaches infinity of (sqrt(x^2+2x)-x)^{6/x ln(cos(1/x))} is 1