Calculus Examples

$lim_{x \to 8} \frac{x + \sin (x)}{2 x + 7 - 5 \sin (x)}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} x + \sin (x)}{lim_{x \to 8} 2 x + 7 - 5 \sin (x)}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} x + lim_{x \to 8} \sin (x)}{lim_{x \to 8} 2 x + 7 - 5 \sin (x)}$

Step 3

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

$\frac{lim_{x \to 8} x + \sin (lim_{x \to 8} x)}{lim_{x \to 8} 2 x + 7 - 5 \sin (x)}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} x + \sin (lim_{x \to 8} x)}{lim_{x \to 8} 2 x + lim_{x \to 8} 7 - lim_{x \to 8} 5 \sin (x)}$

Step 5

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

$\frac{lim_{x \to 8} x + \sin (lim_{x \to 8} x)}{2 lim_{x \to 8} x + lim_{x \to 8} 7 - lim_{x \to 8} 5 \sin (x)}$

Step 6

Evaluate the limit of $7$ which is constant as $x$ approaches $8$ .

$\frac{lim_{x \to 8} x + \sin (lim_{x \to 8} x)}{2 lim_{x \to 8} x + 7 - lim_{x \to 8} 5 \sin (x)}$

Step 7

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{lim_{x \to 8} x + \sin (lim_{x \to 8} x)}{2 lim_{x \to 8} x + 7 - 5 lim_{x \to 8} \sin (x)}$

Step 8

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

$\frac{lim_{x \to 8} x + \sin (lim_{x \to 8} x)}{2 lim_{x \to 8} x + 7 - 5 \sin (lim_{x \to 8} x)}$

Step 9

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{8 + \sin (lim_{x \to 8} x)}{2 lim_{x \to 8} x + 7 - 5 \sin (lim_{x \to 8} x)}$

Step 9.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{8 + \sin (8)}{2 lim_{x \to 8} x + 7 - 5 \sin (lim_{x \to 8} x)}$

Step 9.3

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{8 + \sin (8)}{2 \cdot 8 + 7 - 5 \sin (lim_{x \to 8} x)}$

Step 9.4

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{8 + \sin (8)}{2 \cdot 8 + 7 - 5 \sin (8)}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Evaluate $\sin (8)$ .

$\frac{8 + 0.1391731}{2 \cdot 8 + 7 - 5 \sin (8)}$

Step 10.1.2

Add $8$ and $0.1391731$ .

$\frac{8.1391731}{2 \cdot 8 + 7 - 5 \sin (8)}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Multiply $2$ by $8$ .

$\frac{8.1391731}{16 + 7 - 5 \sin (8)}$

Step 10.2.2

Evaluate $\sin (8)$ .

$\frac{8.1391731}{16 + 7 - 5 \cdot 0.1391731}$

Step 10.2.3

Multiply $- 5$ by $0.1391731$ .

$\frac{8.1391731}{16 + 7 - 0.6958655}$

Step 10.2.4

Add $16$ and $7$ .

$\frac{8.1391731}{23 - 0.6958655}$

Step 10.2.5

Subtract $0.6958655$ from $23$ .

$\frac{8.1391731}{22.30413449}$

Step 10.3

Divide $8.1391731$ by $22.30413449$ .

$0.36491768$