Calculus Examples

$lim_{x \to 8} \sqrt{3 x^{2} + 5 x + 3} - \sqrt{16 x^{2} - 3 x + 4}$

Step 1

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

$lim_{x \to 8} \sqrt{3 x^{2} + 5 x + 3} - lim_{x \to 8} \sqrt{16 x^{2} - 3 x + 4}$

Step 2

Move the limit under the radical sign.

$\sqrt{lim_{x \to 8} 3 x^{2} + 5 x + 3} - lim_{x \to 8} \sqrt{16 x^{2} - 3 x + 4}$

Step 3

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

$\sqrt{lim_{x \to 8} 3 x^{2} + lim_{x \to 8} 5 x + lim_{x \to 8} 3} - lim_{x \to 8} \sqrt{16 x^{2} - 3 x + 4}$

Step 4

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

$\sqrt{3 lim_{x \to 8} x^{2} + lim_{x \to 8} 5 x + lim_{x \to 8} 3} - lim_{x \to 8} \sqrt{16 x^{2} - 3 x + 4}$

Step 5

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

$\sqrt{3 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 5 x + lim_{x \to 8} 3} - lim_{x \to 8} \sqrt{16 x^{2} - 3 x + 4}$

Step 6

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

$\sqrt{3 {(lim_{x \to 8} x)}^{2} + 5 lim_{x \to 8} x + lim_{x \to 8} 3} - lim_{x \to 8} \sqrt{16 x^{2} - 3 x + 4}$

Step 7

Evaluate the limit of $3$ which is constant as $x$ approaches $8$ .

$\sqrt{3 {(lim_{x \to 8} x)}^{2} + 5 lim_{x \to 8} x + 3} - lim_{x \to 8} \sqrt{16 x^{2} - 3 x + 4}$

Step 8

Move the limit under the radical sign.

$\sqrt{3 {(lim_{x \to 8} x)}^{2} + 5 lim_{x \to 8} x + 3} - \sqrt{lim_{x \to 8} 16 x^{2} - 3 x + 4}$

Step 9

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

$\sqrt{3 {(lim_{x \to 8} x)}^{2} + 5 lim_{x \to 8} x + 3} - \sqrt{lim_{x \to 8} 16 x^{2} - lim_{x \to 8} 3 x + lim_{x \to 8} 4}$

Step 10

Move the term $16$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{3 {(lim_{x \to 8} x)}^{2} + 5 lim_{x \to 8} x + 3} - \sqrt{16 lim_{x \to 8} x^{2} - lim_{x \to 8} 3 x + lim_{x \to 8} 4}$

Step 11

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

$\sqrt{3 {(lim_{x \to 8} x)}^{2} + 5 lim_{x \to 8} x + 3} - \sqrt{16 {(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 3 x + lim_{x \to 8} 4}$

Step 12

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

$\sqrt{3 {(lim_{x \to 8} x)}^{2} + 5 lim_{x \to 8} x + 3} - \sqrt{16 {(lim_{x \to 8} x)}^{2} - 3 lim_{x \to 8} x + lim_{x \to 8} 4}$

Step 13

Evaluate the limit of $4$ which is constant as $x$ approaches $8$ .

$\sqrt{3 {(lim_{x \to 8} x)}^{2} + 5 lim_{x \to 8} x + 3} - \sqrt{16 {(lim_{x \to 8} x)}^{2} - 3 lim_{x \to 8} x + 4}$

Step 14

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

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Step 14.1

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

$\sqrt{3 \cdot 8^{2} + 5 lim_{x \to 8} x + 3} - \sqrt{16 {(lim_{x \to 8} x)}^{2} - 3 lim_{x \to 8} x + 4}$

Step 14.2

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

$\sqrt{3 \cdot 8^{2} + 5 \cdot 8 + 3} - \sqrt{16 {(lim_{x \to 8} x)}^{2} - 3 lim_{x \to 8} x + 4}$

Step 14.3

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

$\sqrt{3 \cdot 8^{2} + 5 \cdot 8 + 3} - \sqrt{16 \cdot 8^{2} - 3 lim_{x \to 8} x + 4}$

Step 14.4

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

$\sqrt{3 \cdot 8^{2} + 5 \cdot 8 + 3} - \sqrt{16 \cdot 8^{2} - 3 \cdot 8 + 4}$

Step 15

Simplify each term.

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Step 15.1

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

$\sqrt{3 \cdot 64 + 5 \cdot 8 + 3} - \sqrt{16 \cdot 8^{2} - 3 \cdot 8 + 4}$

Step 15.2

Multiply $3$ by $64$ .

$\sqrt{192 + 5 \cdot 8 + 3} - \sqrt{16 \cdot 8^{2} - 3 \cdot 8 + 4}$

Step 15.3

Multiply $5$ by $8$ .

$\sqrt{192 + 40 + 3} - \sqrt{16 \cdot 8^{2} - 3 \cdot 8 + 4}$

Step 15.4

Add $192$ and $40$ .

$\sqrt{232 + 3} - \sqrt{16 \cdot 8^{2} - 3 \cdot 8 + 4}$

Step 15.5

Add $232$ and $3$ .

$\sqrt{235} - \sqrt{16 \cdot 8^{2} - 3 \cdot 8 + 4}$

Step 15.6

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

$\sqrt{235} - \sqrt{16 \cdot 64 - 3 \cdot 8 + 4}$

Step 15.7

Multiply $16$ by $64$ .

$\sqrt{235} - \sqrt{1024 - 3 \cdot 8 + 4}$

Step 15.8

Multiply $- 3$ by $8$ .

$\sqrt{235} - \sqrt{1024 - 24 + 4}$

Step 15.9

Subtract $24$ from $1024$ .

$\sqrt{235} - \sqrt{1000 + 4}$

Step 15.10

Add $1000$ and $4$ .

$\sqrt{235} - \sqrt{1004}$

Step 15.11

Rewrite $1004$ as $2^{2} \cdot 251$ .

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Step 15.11.1

Factor $4$ out of $1004$ .

$\sqrt{235} - \sqrt{4 (251)}$

Step 15.11.2

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

$\sqrt{235} - \sqrt{2^{2} \cdot 251}$

Step 15.12

Pull terms out from under the radical.

$\sqrt{235} - (2 \sqrt{251})$

Step 15.13

Multiply $2$ by $- 1$ .

$\sqrt{235} - 2 \sqrt{251}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$\sqrt{235} - 2 \sqrt{251}$

Decimal Form:

$- 16.35624931 \dots$