Question

1) EVALUATE

Limit {(2+x)^5/4-(2)^5/4}/{(2+x)^2/3-(2)^2/3}
x tends to zero

Answer 1

Answer on Question #85213 – Math – Real Analysis

Question

1) EVALUATE

Limit $\{(2 + x)^5 / 4 - (2)^5 / 4\} / \{(2 + x)^2 / 3 - (2)^2 / 3\}$

x tends to zero

Solution

$$\lim_{x \to 0} \frac{\left((2 + x)^{\frac{5}{4}} - 2^{\frac{5}{4}}\right)}{(2 + x)^{\frac{2}{3}} - 2^{\frac{2}{3}}} = \binom{0}{0} = \lim_{x \to 0} \frac{\frac{5}{4}(2 + x)^{\frac{1}{4}} - 0}{\frac{2}{3}(2 + x)^{-\frac{1}{3}} - 0} = \frac{15}{8} \cdot \frac{2^{\frac{1}{4}}}{2^{-\frac{1}{3}}} = \frac{15}{8} \cdot 2^{\frac{7}{12}} = \frac{15}{\frac{29}{2^{\frac{1}{12}}}}.$$

Answer: $\lim_{x \to 0} \frac{(2 + x)^{\frac{5}{4}} - 2^{\frac{5}{4}}}{(2 + x)^{\frac{2}{3}} - 2^{\frac{2}{3}}} = \frac{15}{\frac{29}{2^{\frac{1}{12}}}}.$

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