Calculus Examples

$lim_{x \to - 3} \frac{e^{x + 3} - x}{3 \sin (x + 3) + 3 x}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 3$ .

$\frac{lim_{x \to - 3} e^{x + 3} - x}{lim_{x \to - 3} 3 \sin (x + 3) + 3 x}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

$\frac{lim_{x \to - 3} e^{x + 3} - lim_{x \to - 3} x}{lim_{x \to - 3} 3 \sin (x + 3) + 3 x}$

Step 3

Move the limit into the exponent.

$\frac{e^{lim_{x \to - 3} x + 3} - lim_{x \to - 3} x}{lim_{x \to - 3} 3 \sin (x + 3) + 3 x}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

$\frac{e^{lim_{x \to - 3} x + lim_{x \to - 3} 3} - lim_{x \to - 3} x}{lim_{x \to - 3} 3 \sin (x + 3) + 3 x}$

Step 5

Evaluate the limit of $3$ which is constant as $x$ approaches $- 3$ .

$\frac{e^{lim_{x \to - 3} x + 3} - lim_{x \to - 3} x}{lim_{x \to - 3} 3 \sin (x + 3) + 3 x}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

$\frac{e^{lim_{x \to - 3} x + 3} - lim_{x \to - 3} x}{lim_{x \to - 3} 3 \sin (x + 3) + lim_{x \to - 3} 3 x}$

Step 7

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

$\frac{e^{lim_{x \to - 3} x + 3} - lim_{x \to - 3} x}{3 lim_{x \to - 3} \sin (x + 3) + lim_{x \to - 3} 3 x}$

Step 8

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

$\frac{e^{lim_{x \to - 3} x + 3} - lim_{x \to - 3} x}{3 \sin (lim_{x \to - 3} x + 3) + lim_{x \to - 3} 3 x}$

Step 9

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

$\frac{e^{lim_{x \to - 3} x + 3} - lim_{x \to - 3} x}{3 \sin (lim_{x \to - 3} x + lim_{x \to - 3} 3) + lim_{x \to - 3} 3 x}$

Step 10

Evaluate the limit of $3$ which is constant as $x$ approaches $- 3$ .

$\frac{e^{lim_{x \to - 3} x + 3} - lim_{x \to - 3} x}{3 \sin (lim_{x \to - 3} x + 3) + lim_{x \to - 3} 3 x}$

Step 11

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

$\frac{e^{lim_{x \to - 3} x + 3} - lim_{x \to - 3} x}{3 \sin (lim_{x \to - 3} x + 3) + 3 lim_{x \to - 3} x}$

Step 12

Evaluate the limits by plugging in $- 3$ for all occurrences of $x$ .

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Step 12.1

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$\frac{e^{- 3 + 3} - lim_{x \to - 3} x}{3 \sin (lim_{x \to - 3} x + 3) + 3 lim_{x \to - 3} x}$

Step 12.2

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$\frac{e^{- 3 + 3} + 3}{3 \sin (lim_{x \to - 3} x + 3) + 3 lim_{x \to - 3} x}$

Step 12.3

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$\frac{e^{- 3 + 3} + 3}{3 \sin (- 3 + 3) + 3 lim_{x \to - 3} x}$

Step 12.4

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$\frac{e^{- 3 + 3} + 3}{3 \sin (- 3 + 3) + 3 \cdot - 3}$

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Add $- 3$ and $3$ .

$\frac{e^{0} + 3}{3 \sin (- 3 + 3) + 3 \cdot - 3}$

Step 13.1.2

Anything raised to $0$ is $1$ .

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

Step 13.1.3

Add $1$ and $3$ .

$\frac{4}{3 \sin (- 3 + 3) + 3 \cdot - 3}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Add $- 3$ and $3$ .

$\frac{4}{3 \sin (0) + 3 \cdot - 3}$

Step 13.2.2

The exact value of $\sin (0)$ is $0$ .

$\frac{4}{3 \cdot 0 + 3 \cdot - 3}$

Step 13.2.3

Multiply $3$ by $0$ .

$\frac{4}{0 + 3 \cdot - 3}$

Step 13.2.4

Multiply $3$ by $- 3$ .

$\frac{4}{0 - 9}$

Step 13.2.5

Subtract $9$ from $0$ .

$\frac{4}{- 9}$

Step 13.3

Move the negative in front of the fraction.

$- \frac{4}{9}$