Calculus Examples

$lim_{x \to 9} \frac{5 \sin (7 x - 35)}{5 (x - 5)}$

Step 1

Cancel the common factor of $5$ .

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Step 1.1

Cancel the common factor.

$lim_{x \to 9} \frac{5 \sin (7 x - 35)}{5 (x - 5)}$

Step 1.2

Rewrite the expression.

$lim_{x \to 9} \frac{\sin (7 x - 35)}{x - 5}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $9$ .

$\frac{lim_{x \to 9} \sin (7 x - 35)}{lim_{x \to 9} x - 5}$

Step 3

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

$\frac{\sin (lim_{x \to 9} 7 x - 35)}{lim_{x \to 9} x - 5}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $9$ .

$\frac{\sin (lim_{x \to 9} 7 x - lim_{x \to 9} 35)}{lim_{x \to 9} x - 5}$

Step 5

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sin (7 lim_{x \to 9} x - lim_{x \to 9} 35)}{lim_{x \to 9} x - 5}$

Step 6

Evaluate the limit of $35$ which is constant as $x$ approaches $9$ .

$\frac{\sin (7 lim_{x \to 9} x - 1 \cdot 35)}{lim_{x \to 9} x - 5}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $9$ .

$\frac{\sin (7 lim_{x \to 9} x - 1 \cdot 35)}{lim_{x \to 9} x - lim_{x \to 9} 5}$

Step 8

Evaluate the limit of $5$ which is constant as $x$ approaches $9$ .

$\frac{\sin (7 lim_{x \to 9} x - 1 \cdot 35)}{lim_{x \to 9} x - 1 \cdot 5}$

Step 9

Evaluate the limits by plugging in $9$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$\frac{\sin (7 \cdot 9 - 1 \cdot 35)}{lim_{x \to 9} x - 1 \cdot 5}$

Step 9.2

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$\frac{\sin (7 \cdot 9 - 1 \cdot 35)}{9 - 1 \cdot 5}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Simplify each term.

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Step 10.1.1.1

Multiply $7$ by $9$ .

$\frac{\sin (63 - 1 \cdot 35)}{9 - 1 \cdot 5}$

Step 10.1.1.2

Multiply $- 1$ by $35$ .

$\frac{\sin (63 - 35)}{9 - 1 \cdot 5}$

Step 10.1.2

Subtract $35$ from $63$ .

$\frac{\sin (28)}{9 - 1 \cdot 5}$

Step 10.1.3

Evaluate $\sin (28)$ .

$\frac{0.46947156}{9 - 1 \cdot 5}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Multiply $- 1$ by $5$ .

$\frac{0.46947156}{9 - 5}$

Step 10.2.2

Subtract $5$ from $9$ .

$\frac{0.46947156}{4}$

Step 10.3

Divide $0.46947156$ by $4$ .

$0.11736789$