Calculus Examples

$lim_{x \to 0} e^{4 x} - 2 e^{2 x} + 1$

Step 1

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

$lim_{x \to 0} e^{4 x} - lim_{x \to 0} 2 e^{2 x} + lim_{x \to 0} 1$

Step 2

Move the limit into the exponent.

$e^{lim_{x \to 0} 4 x} - lim_{x \to 0} 2 e^{2 x} + lim_{x \to 0} 1$

Step 3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$e^{4 lim_{x \to 0} x} - lim_{x \to 0} 2 e^{2 x} + lim_{x \to 0} 1$

Step 4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$e^{4 lim_{x \to 0} x} - 2 lim_{x \to 0} e^{2 x} + lim_{x \to 0} 1$

Step 5

Move the limit into the exponent.

$e^{4 lim_{x \to 0} x} - 2 e^{lim_{x \to 0} 2 x} + lim_{x \to 0} 1$

Step 6

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$e^{4 lim_{x \to 0} x} - 2 e^{2 lim_{x \to 0} x} + lim_{x \to 0} 1$

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$e^{4 lim_{x \to 0} x} - 2 e^{2 lim_{x \to 0} x} + 1$

Step 8

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

Tap for more steps...

Step 8.1

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

$e^{4 \cdot 0} - 2 e^{2 lim_{x \to 0} x} + 1$

Step 8.2

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

$e^{4 \cdot 0} - 2 e^{2 \cdot 0} + 1$

Step 9

Simplify the answer.

Tap for more steps...

Step 9.1

Simplify each term.

Tap for more steps...

Step 9.1.1

Multiply $4$ by $0$ .

$e^{0} - 2 e^{2 \cdot 0} + 1$

Step 9.1.2

Anything raised to $0$ is $1$ .

$1 - 2 e^{2 \cdot 0} + 1$

Step 9.1.3

Multiply $2$ by $0$ .

$1 - 2 e^{0} + 1$

Step 9.1.4

Anything raised to $0$ is $1$ .

$1 - 2 \cdot 1 + 1$

Step 9.1.5

Multiply $- 2$ by $1$ .

$1 - 2 + 1$

Step 9.2

Subtract $2$ from $1$ .

$- 1 + 1$

Step 9.3

Add $- 1$ and $1$ .

$0$