Calculus Examples

$lim_{x \to 8} \frac{1}{2 x - \sqrt{4 x^{2} - 3 x + 2}}$

Step 1

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

$\frac{lim_{x \to 8} 1}{lim_{x \to 8} 2 x - \sqrt{4 x^{2} - 3 x + 2}}$

Step 2

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

$\frac{1}{lim_{x \to 8} 2 x - \sqrt{4 x^{2} - 3 x + 2}}$

Step 3

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

$\frac{1}{lim_{x \to 8} 2 x - lim_{x \to 8} \sqrt{4 x^{2} - 3 x + 2}}$

Step 4

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

$\frac{1}{2 lim_{x \to 8} x - lim_{x \to 8} \sqrt{4 x^{2} - 3 x + 2}}$

Step 5

Move the limit under the radical sign.

$\frac{1}{2 lim_{x \to 8} x - \sqrt{lim_{x \to 8} 4 x^{2} - 3 x + 2}}$

Step 6

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

$\frac{1}{2 lim_{x \to 8} x - \sqrt{lim_{x \to 8} 4 x^{2} - lim_{x \to 8} 3 x + lim_{x \to 8} 2}}$

Step 7

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2 lim_{x \to 8} x - \sqrt{4 lim_{x \to 8} x^{2} - lim_{x \to 8} 3 x + lim_{x \to 8} 2}}$

Step 8

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

$\frac{1}{2 lim_{x \to 8} x - \sqrt{4 {(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 3 x + lim_{x \to 8} 2}}$

Step 9

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

$\frac{1}{2 lim_{x \to 8} x - \sqrt{4 {(lim_{x \to 8} x)}^{2} - 3 lim_{x \to 8} x + lim_{x \to 8} 2}}$

Step 10

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$\frac{1}{2 lim_{x \to 8} x - \sqrt{4 {(lim_{x \to 8} x)}^{2} - 3 lim_{x \to 8} x + 2}}$

Step 11

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

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Step 11.1

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

$\frac{1}{2 \cdot 8 - \sqrt{4 {(lim_{x \to 8} x)}^{2} - 3 lim_{x \to 8} x + 2}}$

Step 11.2

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

$\frac{1}{2 \cdot 8 - \sqrt{4 \cdot 8^{2} - 3 lim_{x \to 8} x + 2}}$

Step 11.3

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

$\frac{1}{2 \cdot 8 - \sqrt{4 \cdot 8^{2} - 3 \cdot 8 + 2}}$

Step 12

Simplify the answer.

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Step 12.1

Simplify the denominator.

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Step 12.1.1

Multiply $2$ by $8$ .

$\frac{1}{16 - \sqrt{4 \cdot 8^{2} - 3 \cdot 8 + 2}}$

Step 12.1.2

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

$\frac{1}{16 - \sqrt{4 \cdot 64 - 3 \cdot 8 + 2}}$

Step 12.1.3

Multiply $4$ by $64$ .

$\frac{1}{16 - \sqrt{256 - 3 \cdot 8 + 2}}$

Step 12.1.4

Multiply $- 3$ by $8$ .

$\frac{1}{16 - \sqrt{256 - 24 + 2}}$

Step 12.1.5

Subtract $24$ from $256$ .

$\frac{1}{16 - \sqrt{232 + 2}}$

Step 12.1.6

Add $232$ and $2$ .

$\frac{1}{16 - \sqrt{234}}$

Step 12.1.7

Rewrite $234$ as $3^{2} \cdot 26$ .

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Step 12.1.7.1

Factor $9$ out of $234$ .

$\frac{1}{16 - \sqrt{9 (26)}}$

Step 12.1.7.2

Rewrite $9$ as $3^{2}$ .

$\frac{1}{16 - \sqrt{3^{2} \cdot 26}}$

Step 12.1.8

Pull terms out from under the radical.

$\frac{1}{16 - (3 \sqrt{26})}$

Step 12.1.9

Multiply $3$ by $- 1$ .

$\frac{1}{16 - 3 \sqrt{26}}$

Step 12.2

Multiply $\frac{1}{16 - 3 \sqrt{26}}$ by $\frac{16 + 3 \sqrt{26}}{16 + 3 \sqrt{26}}$ .

$\frac{1}{16 - 3 \sqrt{26}} \cdot \frac{16 + 3 \sqrt{26}}{16 + 3 \sqrt{26}}$

Step 12.3

Multiply $\frac{1}{16 - 3 \sqrt{26}}$ by $\frac{16 + 3 \sqrt{26}}{16 + 3 \sqrt{26}}$ .

$\frac{16 + 3 \sqrt{26}}{(16 - 3 \sqrt{26}) (16 + 3 \sqrt{26})}$

Step 12.4

Expand the denominator using the FOIL method.

$\frac{16 + 3 \sqrt{26}}{256 + 48 \sqrt{26} - 48 \sqrt{26} - 9 {\sqrt{26}}^{2}}$

Step 12.5

Simplify.

$\frac{16 + 3 \sqrt{26}}{22}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{16 + 3 \sqrt{26}}{22}$

Decimal Form:

$1.42259357 \dots$