Calculus Examples

$lim_{x \to (- 1)} \frac{x^{2} + 3 x}{2 \tan (x)}$

Step 1

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} lim_{x \to (- 1)} \frac{x^{2} + 3 x}{\tan (x)}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $(- 1)$ .

$\frac{1}{2} \cdot \frac{lim_{x \to (- 1)} x^{2} + 3 x}{lim_{x \to (- 1)} \tan (x)}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $(- 1)$ .

$\frac{1}{2} \cdot \frac{lim_{x \to (- 1)} x^{2} + lim_{x \to (- 1)} 3 x}{lim_{x \to (- 1)} \tan (x)}$

Step 4

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

$\frac{1}{2} \cdot \frac{{(lim_{x \to (- 1)} x)}^{2} + lim_{x \to (- 1)} 3 x}{lim_{x \to (- 1)} \tan (x)}$

Step 5

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} \cdot \frac{{(lim_{x \to (- 1)} x)}^{2} + 3 lim_{x \to (- 1)} x}{lim_{x \to (- 1)} \tan (x)}$

Step 6

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

$\frac{1}{2} \cdot \frac{{(lim_{x \to (- 1)} x)}^{2} + 3 lim_{x \to (- 1)} x}{\tan (lim_{x \to (- 1)} x)}$

Step 7

Evaluate the limits by plugging in $(- 1)$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $(- 1)$ for $x$ .

$\frac{1}{2} \cdot \frac{{(- 1)}^{2} + 3 lim_{x \to (- 1)} x}{\tan (lim_{x \to (- 1)} x)}$

Step 7.2

Evaluate the limit of $x$ by plugging in $(- 1)$ for $x$ .

$\frac{1}{2} \cdot \frac{{(- 1)}^{2} + 3 (- 1)}{\tan (lim_{x \to (- 1)} x)}$

Step 7.3

Evaluate the limit of $x$ by plugging in $(- 1)$ for $x$ .

$\frac{1}{2} \cdot \frac{{(- 1)}^{2} + 3 (- 1)}{\tan ((- 1))}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 1$ to the power of $2$ .

$\frac{1}{2} \cdot \frac{1 + 3 \cdot - 1}{\tan (- 1)}$

Step 8.1.2

Multiply $3$ by $- 1$ .

$\frac{1}{2} \cdot \frac{1 - 3}{\tan (- 1)}$

Step 8.1.3

Subtract $3$ from $1$ .

$\frac{1}{2} \cdot \frac{- 2}{\tan (- 1)}$

Step 8.2

Evaluate $\tan (- 1)$ .

$\frac{1}{2} \cdot \frac{- 2}{- 1.55740772}$

Step 8.3

Cancel the common factor of $2$ .

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Step 8.3.1

Factor $2$ out of $- 2$ .

$\frac{1}{2} \cdot \frac{2 (- 1)}{- 1.55740772}$

Step 8.3.2

Cancel the common factor.

$\frac{1}{2} \cdot \frac{2 \cdot - 1}{- 1.55740772}$

Step 8.3.3

Rewrite the expression.

$\frac{- 1}{- 1.55740772}$

Step 8.4

Divide $- 1$ by $- 1.55740772$ .

$0.64209261$