What is the limit as x approaches 0 of (e^{ax}-e^{bx})/x ?

The limit as x approaches 0 of (e^{ax}-e^{bx})/x is a-b

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limit as x approaches 0 of (e^{ax}-e^{bx})/x

x \to 0 lim (\frac{e ^{a x} - e ^{b x}}{x})

a - b

Solution steps

x \to 0 lim (\frac{e ^{a x} - e ^{b x}}{x})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{e ^{a x} a - e ^{b x} b}{1})

Plug in the value

x = 0

= \frac{e ^{a \cdot 0} a - e ^{b \cdot 0} b}{1}

Simplify

\frac{e ^{a \cdot 0} a - e ^{b \cdot 0} b}{1} : a - b

= a - b

What is the limit as x approaches 0 of (e^{ax}-e^{bx})/x ?
The limit as x approaches 0 of (e^{ax}-e^{bx})/x is a-b