Calculus Examples

$lim_{t \to 8} \frac{\sqrt{t} + t^{2}}{2 t - t^{2}}$

Step 1

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

$\frac{lim_{t \to 8} \sqrt{t} + t^{2}}{lim_{t \to 8} 2 t - t^{2}}$

Step 2

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

$\frac{lim_{t \to 8} \sqrt{t} + lim_{t \to 8} t^{2}}{lim_{t \to 8} 2 t - t^{2}}$

Step 3

Move the limit under the radical sign.

$\frac{\sqrt{lim_{t \to 8} t} + lim_{t \to 8} t^{2}}{lim_{t \to 8} 2 t - t^{2}}$

Step 4

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

$\frac{\sqrt{lim_{t \to 8} t} + {(lim_{t \to 8} t)}^{2}}{lim_{t \to 8} 2 t - t^{2}}$

Step 5

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

$\frac{\sqrt{lim_{t \to 8} t} + {(lim_{t \to 8} t)}^{2}}{lim_{t \to 8} 2 t - lim_{t \to 8} t^{2}}$

Step 6

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

$\frac{\sqrt{lim_{t \to 8} t} + {(lim_{t \to 8} t)}^{2}}{2 lim_{t \to 8} t - lim_{t \to 8} t^{2}}$

Step 7

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

$\frac{\sqrt{lim_{t \to 8} t} + {(lim_{t \to 8} t)}^{2}}{2 lim_{t \to 8} t - {(lim_{t \to 8} t)}^{2}}$

Step 8

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

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Step 8.1

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

$\frac{\sqrt{8} + {(lim_{t \to 8} t)}^{2}}{2 lim_{t \to 8} t - {(lim_{t \to 8} t)}^{2}}$

Step 8.2

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

$\frac{\sqrt{8} + 8^{2}}{2 lim_{t \to 8} t - {(lim_{t \to 8} t)}^{2}}$

Step 8.3

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

$\frac{\sqrt{8} + 8^{2}}{2 \cdot 8 - {(lim_{t \to 8} t)}^{2}}$

Step 8.4

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

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

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 9.1.1.1

Factor $4$ out of $8$ .

$\frac{\sqrt{4 (2)} + 8^{2}}{2 \cdot 8 - 8^{2}}$

Step 9.1.1.2

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

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

Step 9.1.2

Pull terms out from under the radical.

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

Step 9.1.3

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

$\frac{2 \sqrt{2} + 64}{2 \cdot 8 - 8^{2}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Multiply $2$ by $8$ .

$\frac{2 \sqrt{2} + 64}{16 - 8^{2}}$

Step 9.2.2

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

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

Step 9.2.3

Multiply $- 1$ by $64$ .

$\frac{2 \sqrt{2} + 64}{16 - 64}$

Step 9.2.4

Subtract $64$ from $16$ .

$\frac{2 \sqrt{2} + 64}{- 48}$

Step 9.3

Cancel the common factor of $2 \sqrt{2} + 64$ and $- 48$ .

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Step 9.3.1

Factor $2$ out of $2 \sqrt{2}$ .

$\frac{2 (\sqrt{2}) + 64}{- 48}$

Step 9.3.2

Factor $2$ out of $64$ .

$\frac{2 \sqrt{2} + 2 \cdot 32}{- 48}$

Step 9.3.3

Factor $2$ out of $2 \sqrt{2} + 2 \cdot 32$ .

$\frac{2 (\sqrt{2} + 32)}{- 48}$

Step 9.3.4

Cancel the common factors.

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Step 9.3.4.1

Factor $2$ out of $- 48$ .

$\frac{2 (\sqrt{2} + 32)}{2 (- 24)}$

Step 9.3.4.2

Cancel the common factor.

$\frac{2 (\sqrt{2} + 32)}{2 \cdot - 24}$

Step 9.3.4.3

Rewrite the expression.

$\frac{\sqrt{2} + 32}{- 24}$

Step 9.4

Move the negative in front of the fraction.

$- \frac{\sqrt{2} + 32}{24}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{2} + 32}{24}$

Decimal Form:

$- 1.39225889 \dots$