Calculus Examples

$lim_{x \to 3} \frac{\sin (x - 3) + 14 x + 4}{\sqrt{x^{2} - 5}}$

Step 1

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

$\frac{lim_{x \to 3} \sin (x - 3) + 14 x + 4}{lim_{x \to 3} \sqrt{x^{2} - 5}}$

Step 2

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

$\frac{lim_{x \to 3} \sin (x - 3) + lim_{x \to 3} 14 x + lim_{x \to 3} 4}{lim_{x \to 3} \sqrt{x^{2} - 5}}$

Step 3

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

$\frac{\sin (lim_{x \to 3} x - 3) + lim_{x \to 3} 14 x + lim_{x \to 3} 4}{lim_{x \to 3} \sqrt{x^{2} - 5}}$

Step 4

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

$\frac{\sin (lim_{x \to 3} x - lim_{x \to 3} 3) + lim_{x \to 3} 14 x + lim_{x \to 3} 4}{lim_{x \to 3} \sqrt{x^{2} - 5}}$

Step 5

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

$\frac{\sin (lim_{x \to 3} x - 1 \cdot 3) + lim_{x \to 3} 14 x + lim_{x \to 3} 4}{lim_{x \to 3} \sqrt{x^{2} - 5}}$

Step 6

Move the term $14$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sin (lim_{x \to 3} x - 1 \cdot 3) + 14 lim_{x \to 3} x + lim_{x \to 3} 4}{lim_{x \to 3} \sqrt{x^{2} - 5}}$

Step 7

Evaluate the limit of $4$ which is constant as $x$ approaches $3$ .

$\frac{\sin (lim_{x \to 3} x - 1 \cdot 3) + 14 lim_{x \to 3} x + 4}{lim_{x \to 3} \sqrt{x^{2} - 5}}$

Step 8

Move the limit under the radical sign.

$\frac{\sin (lim_{x \to 3} x - 1 \cdot 3) + 14 lim_{x \to 3} x + 4}{\sqrt{lim_{x \to 3} x^{2} - 5}}$

Step 9

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

$\frac{\sin (lim_{x \to 3} x - 1 \cdot 3) + 14 lim_{x \to 3} x + 4}{\sqrt{lim_{x \to 3} x^{2} - lim_{x \to 3} 5}}$

Step 10

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

$\frac{\sin (lim_{x \to 3} x - 1 \cdot 3) + 14 lim_{x \to 3} x + 4}{\sqrt{{(lim_{x \to 3} x)}^{2} - lim_{x \to 3} 5}}$

Step 11

Evaluate the limit of $5$ which is constant as $x$ approaches $3$ .

$\frac{\sin (lim_{x \to 3} x - 1 \cdot 3) + 14 lim_{x \to 3} x + 4}{\sqrt{{(lim_{x \to 3} x)}^{2} - 1 \cdot 5}}$

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 $x$ .

$\frac{\sin (3 - 1 \cdot 3) + 14 lim_{x \to 3} x + 4}{\sqrt{{(lim_{x \to 3} x)}^{2} - 1 \cdot 5}}$

Step 12.2

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

$\frac{\sin (3 - 1 \cdot 3) + 14 \cdot 3 + 4}{\sqrt{{(lim_{x \to 3} x)}^{2} - 1 \cdot 5}}$

Step 12.3

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

$\frac{\sin (3 - 1 \cdot 3) + 14 \cdot 3 + 4}{\sqrt{3^{2} - 1 \cdot 5}}$

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Multiply $- 1$ by $3$ .

$\frac{\sin (3 - 3) + 14 \cdot 3 + 4}{\sqrt{3^{2} - 1 \cdot 5}}$

Step 13.1.2

Subtract $3$ from $3$ .

$\frac{\sin (0) + 14 \cdot 3 + 4}{\sqrt{3^{2} - 1 \cdot 5}}$

Step 13.1.3

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

$\frac{0 + 14 \cdot 3 + 4}{\sqrt{3^{2} - 1 \cdot 5}}$

Step 13.1.4

Multiply $14$ by $3$ .

$\frac{0 + 42 + 4}{\sqrt{3^{2} - 1 \cdot 5}}$

Step 13.1.5

Add $0$ and $42$ .

$\frac{42 + 4}{\sqrt{3^{2} - 1 \cdot 5}}$

Step 13.1.6

Add $42$ and $4$ .

$\frac{46}{\sqrt{3^{2} - 1 \cdot 5}}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Raise $3$ to the power of $2$ .

$\frac{46}{\sqrt{9 - 1 \cdot 5}}$

Step 13.2.2

Multiply $- 1$ by $5$ .

$\frac{46}{\sqrt{9 - 5}}$

Step 13.2.3

Subtract $5$ from $9$ .

$\frac{46}{\sqrt{4}}$

Step 13.2.4

Rewrite $4$ as $2^{2}$ .

$\frac{46}{\sqrt{2^{2}}}$

Step 13.2.5

Pull terms out from under the radical, assuming positive real numbers.

$\frac{46}{2}$

Step 13.3

Divide $46$ by $2$ .

$23$