What is the limit as x approaches 0+of (2^x)/(3^x-1) ?

The limit as x approaches 0+of (2^x)/(3^x-1) is infinity

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limit as x approaches 0+of (2^x)/(3^x-1)

x \to 0 + lim (\frac{2 ^{x}}{3 ^{x} - 1})

\infty

Solution steps

x \to 0 + lim (\frac{2 ^{x}}{3 ^{x} - 1})

\frac{2 ^{x}}{3 ^{x} - 1} = 2^{x} \cdot \frac{1}{3 ^{x} - 1}

= x \to 0 + lim (2^{x} \cdot \frac{1}{3 ^{x} - 1})

x \to a lim [f (x) \cdot g (x)] = x \to a lim f (x) \cdot x \to a lim g (x)

With the exception of indeterminate form

= x \to 0 + lim (2^{x}) \cdot x \to 0 + lim (\frac{1}{3 ^{x} - 1})

x \to 0 + lim (2^{x}) = 1

x \to 0 + lim (\frac{1}{3 ^{x} - 1}) = \infty

= 1 \cdot \infty

Apply Infinity Property:

c \cdot \infty = \infty

= \infty

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What is the limit as x approaches 0+of (2^x)/(3^x-1) ?
The limit as x approaches 0+of (2^x)/(3^x-1) is infinity