Calculus Examples

$lim_{x \to - 8} \frac{\sqrt{49 x^{2} + 1}}{3 x}$

Step 1

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} lim_{x \to - 8} \frac{\sqrt{49 x^{2} + 1}}{x}$

Step 2

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

$\frac{1}{3} \cdot \frac{lim_{x \to - 8} \sqrt{49 x^{2} + 1}}{lim_{x \to - 8} x}$

Step 3

Move the limit under the radical sign.

$\frac{1}{3} \cdot \frac{\sqrt{lim_{x \to - 8} 49 x^{2} + 1}}{lim_{x \to - 8} x}$

Step 4

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

$\frac{1}{3} \cdot \frac{\sqrt{lim_{x \to - 8} 49 x^{2} + lim_{x \to - 8} 1}}{lim_{x \to - 8} x}$

Step 5

Move the term $49$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} \cdot \frac{\sqrt{49 lim_{x \to - 8} x^{2} + lim_{x \to - 8} 1}}{lim_{x \to - 8} x}$

Step 6

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

$\frac{1}{3} \cdot \frac{\sqrt{49 {(lim_{x \to - 8} x)}^{2} + lim_{x \to - 8} 1}}{lim_{x \to - 8} x}$

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $- 8$ .

$\frac{1}{3} \cdot \frac{\sqrt{49 {(lim_{x \to - 8} x)}^{2} + 1}}{lim_{x \to - 8} x}$

Step 8

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

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Step 8.1

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

$\frac{1}{3} \cdot \frac{\sqrt{49 {(- 8)}^{2} + 1}}{lim_{x \to - 8} x}$

Step 8.2

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

$\frac{1}{3} \cdot \frac{\sqrt{49 {(- 8)}^{2} + 1}}{- 8}$

Step 9

Simplify the answer.

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Step 9.1

Combine.

$\frac{1 \sqrt{49 {(- 8)}^{2} + 1}}{3 \cdot - 8}$

Step 9.2

Multiply $\sqrt{49 {(- 8)}^{2} + 1}$ by $1$ .

$\frac{\sqrt{49 {(- 8)}^{2} + 1}}{3 \cdot - 8}$

Step 9.3

Multiply $3$ by $- 8$ .

$\frac{\sqrt{49 {(- 8)}^{2} + 1}}{- 24}$

Step 9.4

Simplify the numerator.

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Step 9.4.1

Raise $- 8$ to the power of $2$ .

$\frac{\sqrt{49 \cdot 64 + 1}}{- 24}$

Step 9.4.2

Multiply $49$ by $64$ .

$\frac{\sqrt{3136 + 1}}{- 24}$

Step 9.4.3

Add $3136$ and $1$ .

$\frac{\sqrt{3137}}{- 24}$

Step 9.5

Move the negative in front of the fraction.

$- \frac{\sqrt{3137}}{24}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{3137}}{24}$

Decimal Form:

$- 2.33370532 \dots$