Calculus Examples

$\frac{lim_{t \to 8} t - t \sqrt{t}}{2 t^{\frac{3}{2}} + 3 t - 5}$

Step 1

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

$\frac{lim_{t \to 8} t - lim_{t \to 8} t \sqrt{t}}{2 t^{\frac{3}{2}} + 3 t - 5}$

Step 2

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

$\frac{lim_{t \to 8} t - lim_{t \to 8} t \cdot lim_{t \to 8} \sqrt{t}}{2 t^{\frac{3}{2}} + 3 t - 5}$

Step 3

Move the limit under the radical sign.

$\frac{lim_{t \to 8} t - lim_{t \to 8} t \cdot \sqrt{lim_{t \to 8} t}}{2 t^{\frac{3}{2}} + 3 t - 5}$

Step 4

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

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Step 4.1

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

$\frac{8 - lim_{t \to 8} t \cdot \sqrt{lim_{t \to 8} t}}{2 t^{\frac{3}{2}} + 3 t - 5}$

Step 4.2

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

$\frac{8 - 8 \cdot \sqrt{lim_{t \to 8} t}}{2 t^{\frac{3}{2}} + 3 t - 5}$

Step 4.3

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

$\frac{8 - 8 \cdot \sqrt{8}}{2 t^{\frac{3}{2}} + 3 t - 5}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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Step 5.1.1.1

Factor $4$ out of $8$ .

$\frac{8 - 8 \cdot \sqrt{4 (2)}}{2 t^{\frac{3}{2}} + 3 t - 5}$

Step 5.1.1.2

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

$\frac{8 - 8 \cdot \sqrt{2^{2} \cdot 2}}{2 t^{\frac{3}{2}} + 3 t - 5}$

Step 5.1.2

Pull terms out from under the radical.

$\frac{8 - 8 \cdot (2 \sqrt{2})}{2 t^{\frac{3}{2}} + 3 t - 5}$

Step 5.1.3

Multiply $2$ by $- 8$ .

$\frac{8 - 16 \sqrt{2}}{2 t^{\frac{3}{2}} + 3 t - 5}$

Step 5.2

Reorder terms.

$\frac{8 - 16 \sqrt{2}}{3 t + 2 t^{\frac{3}{2}} - 5}$