Calculus Examples

$lim_{x \to 0} 2 \cos (x) + \frac{x - 4}{x^{2} + 1}$

Step 1

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

$lim_{x \to 0} 2 \cos (x) + lim_{x \to 0} \frac{x - 4}{x^{2} + 1}$

Step 2

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

$2 lim_{x \to 0} \cos (x) + lim_{x \to 0} \frac{x - 4}{x^{2} + 1}$

Step 3

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

$2 \cos (lim_{x \to 0} x) + lim_{x \to 0} \frac{x - 4}{x^{2} + 1}$

Step 4

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

$2 \cos (lim_{x \to 0} x) + \frac{lim_{x \to 0} x - 4}{lim_{x \to 0} x^{2} + 1}$

Step 5

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

$2 \cos (lim_{x \to 0} x) + \frac{lim_{x \to 0} x - lim_{x \to 0} 4}{lim_{x \to 0} x^{2} + 1}$

Step 6

Evaluate the limit of $4$ which is constant as $x$ approaches $0$ .

$2 \cos (lim_{x \to 0} x) + \frac{lim_{x \to 0} x - 1 \cdot 4}{lim_{x \to 0} x^{2} + 1}$

Step 7

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

$2 \cos (lim_{x \to 0} x) + \frac{lim_{x \to 0} x - 1 \cdot 4}{lim_{x \to 0} x^{2} + lim_{x \to 0} 1}$

Step 8

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

$2 \cos (lim_{x \to 0} x) + \frac{lim_{x \to 0} x - 1 \cdot 4}{{(lim_{x \to 0} x)}^{2} + lim_{x \to 0} 1}$

Step 9

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

$2 \cos (lim_{x \to 0} x) + \frac{lim_{x \to 0} x - 1 \cdot 4}{{(lim_{x \to 0} x)}^{2} + 1}$

Step 10

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

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Step 10.1

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

$2 \cos (0) + \frac{lim_{x \to 0} x - 1 \cdot 4}{{(lim_{x \to 0} x)}^{2} + 1}$

Step 10.2

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

$2 \cos (0) + \frac{0 - 1 \cdot 4}{{(lim_{x \to 0} x)}^{2} + 1}$

Step 10.3

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

$2 \cos (0) + \frac{0 - 1 \cdot 4}{0^{2} + 1}$

Step 11

Simplify the answer.

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Step 11.1

Multiply $- 1$ by $4$ .

$2 \cos (0) + \frac{0 - 4}{0^{2} + 1}$

Step 11.2

Subtract $4$ from $0$ .

$2 \cos (0) + \frac{- 4}{0^{2} + 1}$

Step 11.3

Simplify each term.

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Step 11.3.1

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

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

Step 11.3.2

Multiply $2$ by $1$ .

$2 + \frac{- 4}{0^{2} + 1}$

Step 11.3.3

Simplify the denominator.

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Step 11.3.3.1

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

$2 + \frac{- 4}{0 + 1}$

Step 11.3.3.2

Add $0$ and $1$ .

$2 + \frac{- 4}{1}$

Step 11.3.4

Divide $- 4$ by $1$ .

$2 - 4$

Step 11.4

Subtract $4$ from $2$ .

$- 2$