Calculus Examples

$lim_{t \to 8} \frac{\sqrt{4 t^{2} + 4}}{2 t + 1}$

Step 1

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

$\frac{lim_{t \to 8} \sqrt{4 t^{2} + 4}}{lim_{t \to 8} 2 t + 1}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{t \to 8} 4 t^{2} + 4}}{lim_{t \to 8} 2 t + 1}$

Step 3

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

$\frac{\sqrt{lim_{t \to 8} 4 t^{2} + lim_{t \to 8} 4}}{lim_{t \to 8} 2 t + 1}$

Step 4

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

$\frac{\sqrt{4 lim_{t \to 8} t^{2} + lim_{t \to 8} 4}}{lim_{t \to 8} 2 t + 1}$

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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Step 10.1

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

$\frac{\sqrt{4 \cdot 8^{2} + 4}}{2 lim_{t \to 8} t + 1}$

Step 10.2

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

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

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

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

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

Step 11.1.2

Multiply $4$ by $64$ .

$\frac{\sqrt{256 + 4}}{2 \cdot 8 + 1}$

Step 11.1.3

Add $256$ and $4$ .

$\frac{\sqrt{260}}{2 \cdot 8 + 1}$

Step 11.1.4

Rewrite $260$ as $2^{2} \cdot 65$ .

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Step 11.1.4.1

Factor $4$ out of $260$ .

$\frac{\sqrt{4 (65)}}{2 \cdot 8 + 1}$

Step 11.1.4.2

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

$\frac{\sqrt{2^{2} \cdot 65}}{2 \cdot 8 + 1}$

Step 11.1.5

Pull terms out from under the radical.

$\frac{2 \sqrt{65}}{2 \cdot 8 + 1}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Multiply $2$ by $8$ .

$\frac{2 \sqrt{65}}{16 + 1}$

Step 11.2.2

Add $16$ and $1$ .

$\frac{2 \sqrt{65}}{17}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{65}}{17}$

Decimal Form:

$0.94850091 \dots$