Calculus Examples

$lim_{x \to 0} \frac{1 + 2 e^{2 x} - x^{2}}{3 \cos (π + 2 x)}$

Step 1

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} lim_{x \to 0} \frac{1 + 2 e^{2 x} - x^{2}}{\cos (π + 2 x)}$

Step 2

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

$\frac{1}{3} \cdot \frac{lim_{x \to 0} 1 + 2 e^{2 x} - x^{2}}{lim_{x \to 0} \cos (π + 2 x)}$

Step 3

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

$\frac{1}{3} \cdot \frac{lim_{x \to 0} 1 + lim_{x \to 0} 2 e^{2 x} - lim_{x \to 0} x^{2}}{lim_{x \to 0} \cos (π + 2 x)}$

Step 4

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

$\frac{1}{3} \cdot \frac{1 + lim_{x \to 0} 2 e^{2 x} - lim_{x \to 0} x^{2}}{lim_{x \to 0} \cos (π + 2 x)}$

Step 5

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

$\frac{1}{3} \cdot \frac{1 + 2 lim_{x \to 0} e^{2 x} - lim_{x \to 0} x^{2}}{lim_{x \to 0} \cos (π + 2 x)}$

Step 6

Move the limit into the exponent.

$\frac{1}{3} \cdot \frac{1 + 2 e^{lim_{x \to 0} 2 x} - lim_{x \to 0} x^{2}}{lim_{x \to 0} \cos (π + 2 x)}$

Step 7

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

$\frac{1}{3} \cdot \frac{1 + 2 e^{2 lim_{x \to 0} x} - lim_{x \to 0} x^{2}}{lim_{x \to 0} \cos (π + 2 x)}$

Step 8

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{1}{3} \cdot \frac{1 + 2 e^{2 lim_{x \to 0} x} - {(lim_{x \to 0} x)}^{2}}{lim_{x \to 0} \cos (π + 2 x)}$

Step 9

Move the limit inside the trig function because cosine is continuous.

$\frac{1}{3} \cdot \frac{1 + 2 e^{2 lim_{x \to 0} x} - {(lim_{x \to 0} x)}^{2}}{\cos (lim_{x \to 0} π + 2 x)}$

Step 10

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

$\frac{1}{3} \cdot \frac{1 + 2 e^{2 lim_{x \to 0} x} - {(lim_{x \to 0} x)}^{2}}{\cos (lim_{x \to 0} π + lim_{x \to 0} 2 x)}$

Step 11

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

$\frac{1}{3} \cdot \frac{1 + 2 e^{2 lim_{x \to 0} x} - {(lim_{x \to 0} x)}^{2}}{\cos (π + lim_{x \to 0} 2 x)}$

Step 12

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

$\frac{1}{3} \cdot \frac{1 + 2 e^{2 lim_{x \to 0} x} - {(lim_{x \to 0} x)}^{2}}{\cos (π + 2 lim_{x \to 0} x)}$

Step 13

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

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Step 13.1

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

$\frac{1}{3} \cdot \frac{1 + 2 e^{2 \cdot 0} - {(lim_{x \to 0} x)}^{2}}{\cos (π + 2 lim_{x \to 0} x)}$

Step 13.2

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

$\frac{1}{3} \cdot \frac{1 + 2 e^{2 \cdot 0} - 0^{2}}{\cos (π + 2 lim_{x \to 0} x)}$

Step 13.3

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

$\frac{1}{3} \cdot \frac{1 + 2 e^{2 \cdot 0} - 0^{2}}{\cos (π + 2 \cdot 0)}$

Step 14

Simplify the answer.

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Step 14.1

Simplify the numerator.

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Step 14.1.1

Multiply $2$ by $0$ .

$\frac{1}{3} \cdot \frac{1 + 2 e^{0} - 0^{2}}{\cos (π + 2 \cdot 0)}$

Step 14.1.2

Anything raised to $0$ is $1$ .

$\frac{1}{3} \cdot \frac{1 + 2 \cdot 1 - 0^{2}}{\cos (π + 2 \cdot 0)}$

Step 14.1.3

Multiply $2$ by $1$ .

$\frac{1}{3} \cdot \frac{1 + 2 - 0^{2}}{\cos (π + 2 \cdot 0)}$

Step 14.1.4

Raising $0$ to any positive power yields $0$ .

$\frac{1}{3} \cdot \frac{1 + 2 - 0}{\cos (π + 2 \cdot 0)}$

Step 14.1.5

Multiply $- 1$ by $0$ .

$\frac{1}{3} \cdot \frac{1 + 2 + 0}{\cos (π + 2 \cdot 0)}$

Step 14.1.6

Add $1 + 2$ and $0$ .

$\frac{1}{3} \cdot \frac{1 + 2}{\cos (π + 2 \cdot 0)}$

Step 14.1.7

Add $1$ and $2$ .

$\frac{1}{3} \cdot \frac{3}{\cos (π + 2 \cdot 0)}$

Step 14.2

Simplify the denominator.

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Step 14.2.1

Multiply $2$ by $0$ .

$\frac{1}{3} \cdot \frac{3}{\cos (π + 0)}$

Step 14.2.2

Add $π$ and $0$ .

$\frac{1}{3} \cdot \frac{3}{\cos (π)}$

Step 14.2.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{1}{3} \cdot \frac{3}{- \cos (0)}$

Step 14.2.4

The exact value of $\cos (0)$ is $1$ .

$\frac{1}{3} \cdot \frac{3}{- 1 \cdot 1}$

Step 14.2.5

Multiply $- 1$ by $1$ .

$\frac{1}{3} \cdot \frac{3}{- 1}$

Step 14.3

Cancel the common factor of $3$ .

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Step 14.3.1

Cancel the common factor.

$\frac{1}{3} \cdot \frac{3}{- 1}$

Step 14.3.2

Rewrite the expression.

$\frac{1}{- 1}$

Step 14.4

Divide $1$ by $- 1$ .

$- 1$