Can you find this limit?

Calculus Level 2

lim x 2 2 1 x = ? \large \lim_{x \to \infty} 22^\frac 1x =\ ?

0 1 Positive Infinity Negative Infinity

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2 solutions

As x approaches Infinity, 1/x approaches 0, so 22^(1/x) approaches 1. Here '^' is power notation.

Aram Lindroth
Mar 14, 2018

To make it abundantly obvious, perform a change of variables x 1 y x\to\frac{1}{y} . Then the limit becomes lim y 0 2 2 y \lim_{y\to0} 22^{y} . In this case we can now directly substitute because f ( y ) = 2 2 y f(y)=22^y is defined at y = 0 y=0 : lim y 0 2 2 y = 2 2 0 = 1 \lim_{y\to0} 22^{y} = 22^{0} = 1 .

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