"Solve this questionlimx→∞(x+3)1+1x−x1+1x+3" was answered by students like you

Deegan Chase

Answered question

2022-03-21

Solve this question

lim_{x \to \infty} {(x + 3)}^{1 + \frac{1}{x}} - x^{1 + \frac{1}{x + 3}}

Alannah Farmer

Beginner2022-03-22Added 11 answers

Write the function of interest as

f (x) = \times^{\frac{1}{x}} g (x)

with

g (x) = {(1 + \frac{3}{x})}^{1 + \frac{1}{x}} - x^{\frac{1}{x + 3} - \frac{1}{x}} = e^{a (x)} - e^{b (x)}

with

a (x) = (1 + \frac{1}{x}) \log (1 + \frac{3}{x}) = \frac{3}{x} + o (\frac{1}{x})

and

b (x) = - \frac{3 \log (x)}{x (x + 3)} = o (\frac{1}{x})

This yields

e^{a (x)} = 1 + \frac{3}{x} + o (\frac{1}{x})

and

e^{b (x)} = 1 + o (\frac{1}{x})

hence

g (x) = \frac{3}{x} + o (\frac{1}{x})

Likewise

x^{\frac{1}{x}} = e^{\frac{\log (x)}{x}} = 1 + o (1)

hence

f (x) = \times^{\frac{1}{x}} g (x) = x (1 + o (1)) (\frac{3}{x} + o (\frac{1}{x})) = 3 + o (1)

That is, f does converge and its limit is 3

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