Calculus Examples

$\frac{lim_{h \to 0} e^{x + h} + e^{x}}{h}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{lim_{h \to 0} e^{x + h} + lim_{h \to 0} e^{x}}{h}$

Step 1.2

Move the limit into the exponent.

$\frac{e^{lim_{h \to 0} x + h} + lim_{h \to 0} e^{x}}{h}$

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{e^{lim_{h \to 0} x + lim_{h \to 0} h} + lim_{h \to 0} e^{x}}{h}$

Step 1.4

Evaluate the limit of $x$ which is constant as $h$ approaches $0$ .

$\frac{e^{x + lim_{h \to 0} h} + lim_{h \to 0} e^{x}}{h}$

Step 1.5

Evaluate the limit of $e^{x}$ which is constant as $h$ approaches $0$ .

$\frac{e^{x + lim_{h \to 0} h} + e^{x}}{h}$

Step 2

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{e^{x + 0} + e^{x}}{h}$

Step 3

Simplify the numerator.

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Step 3.1

Add $x$ and $0$ .

$\frac{e^{x} + e^{x}}{h}$

Step 3.2

Add $e^{x}$ and $e^{x}$ .

$\frac{2 e^{x}}{h}$