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

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

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

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

\infty

Solution steps

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

Apply L'Hopital's Rule

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

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

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

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

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

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

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

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

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

= \frac{ln ( 3 )}{2} \cdot 1 \cdot \infty

Apply Infinity Property:

c \cdot \infty = \infty

= \infty

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