Problems from Indeterminate form 1 raised to infinity

Find the following limits:

a) $lim_{x \to + \infty} [\frac{x^{2} + x - 1}{x^{2} + 2}]^{3 x - 1}$

b) $lim_{x \to + \infty} [1 + \frac{3}{2 x}]^{5 x}$

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Development:

a) $lim_{x \to + \infty} [\frac{x^{2} + x - 1}{x^{2} + 2}]^{3 x - 1} = 1^{+ \infty} \Rightarrow e^{lim_{x \to + \infty} [\frac{x^{2} + x - 1}{x^{2} + 2}] (3 x - 1)} =$ $e^{lim_{x \to + \infty} [\frac{x - 3}{x^{2} + 2}] (3 x - 1)} = e^{lim_{x \to + \infty} \frac{3 x^{2} - 10 x + 3}{x^{2} + 2}} = e^{3}$

b) $lim_{x \to + \infty} [1 + \frac{3}{2 x}]^{5 x} = lim_{x \to + \infty} [1 - \frac{3}{2 x}]^{- 5 x} =$ $e^{lim_{x \to + \infty} [1 - \frac{3}{2 x} + 1]^{(- 5 x)}} = e^{lim_{x \to + \infty} \frac{3}{2 x} 5 x} = e^{\frac{3 \cdot 5}{2}} = e^{\frac{15}{2}}$

Solution:

a) $e^{3}$

b) $e^{\frac{15}{2}}$

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