Calculus Examples

$\frac{lim_{t \to 8} 36 t^{3} + 12 t - 653}{7 t^{5} - 13 t^{3} + 4 t^{2}}$

Step 1

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

$\frac{lim_{t \to 8} 36 t^{3} + lim_{t \to 8} 12 t - lim_{t \to 8} 653}{7 t^{5} - 13 t^{3} + 4 t^{2}}$

Step 2

Move the term $36$ outside of the limit because it is constant with respect to $t$ .

$\frac{36 lim_{t \to 8} t^{3} + lim_{t \to 8} 12 t - lim_{t \to 8} 653}{7 t^{5} - 13 t^{3} + 4 t^{2}}$

Step 3

Move the exponent $3$ from $t^{3}$ outside the limit using the Limits Power Rule.

$\frac{36 {(lim_{t \to 8} t)}^{3} + lim_{t \to 8} 12 t - lim_{t \to 8} 653}{7 t^{5} - 13 t^{3} + 4 t^{2}}$

Step 4

Move the term $12$ outside of the limit because it is constant with respect to $t$ .

$\frac{36 {(lim_{t \to 8} t)}^{3} + 12 lim_{t \to 8} t - lim_{t \to 8} 653}{7 t^{5} - 13 t^{3} + 4 t^{2}}$

Step 5

Evaluate the limit of $653$ which is constant as $t$ approaches $8$ .

$\frac{36 {(lim_{t \to 8} t)}^{3} + 12 lim_{t \to 8} t - 1 \cdot 653}{7 t^{5} - 13 t^{3} + 4 t^{2}}$

Step 6

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

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Step 6.1

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

$\frac{36 \cdot 8^{3} + 12 lim_{t \to 8} t - 1 \cdot 653}{7 t^{5} - 13 t^{3} + 4 t^{2}}$

Step 6.2

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

$\frac{36 \cdot 8^{3} + 12 \cdot 8 - 1 \cdot 653}{7 t^{5} - 13 t^{3} + 4 t^{2}}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $8$ to the power of $3$ .

$\frac{36 \cdot 512 + 12 \cdot 8 - 1 \cdot 653}{7 t^{5} - 13 t^{3} + 4 t^{2}}$

Step 7.1.2

Multiply $36$ by $512$ .

$\frac{18432 + 12 \cdot 8 - 1 \cdot 653}{7 t^{5} - 13 t^{3} + 4 t^{2}}$

Step 7.1.3

Multiply $12$ by $8$ .

$\frac{18432 + 96 - 1 \cdot 653}{7 t^{5} - 13 t^{3} + 4 t^{2}}$

Step 7.1.4

Multiply $- 1$ by $653$ .

$\frac{18432 + 96 - 653}{7 t^{5} - 13 t^{3} + 4 t^{2}}$

Step 7.1.5

Add $18432$ and $96$ .

$\frac{18528 - 653}{7 t^{5} - 13 t^{3} + 4 t^{2}}$

Step 7.1.6

Subtract $653$ from $18528$ .

$\frac{17875}{7 t^{5} - 13 t^{3} + 4 t^{2}}$

Step 7.2

Factor $t^{2}$ out of $7 t^{5} - 13 t^{3} + 4 t^{2}$ .

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Step 7.2.1

Factor $t^{2}$ out of $7 t^{5}$ .

$\frac{17875}{t^{2} (7 t^{3}) - 13 t^{3} + 4 t^{2}}$

Step 7.2.2

Factor $t^{2}$ out of $- 13 t^{3}$ .

$\frac{17875}{t^{2} (7 t^{3}) + t^{2} (- 13 t) + 4 t^{2}}$

Step 7.2.3

Factor $t^{2}$ out of $4 t^{2}$ .

$\frac{17875}{t^{2} (7 t^{3}) + t^{2} (- 13 t) + t^{2} \cdot 4}$

Step 7.2.4

Factor $t^{2}$ out of $t^{2} (7 t^{3}) + t^{2} (- 13 t)$ .

$\frac{17875}{t^{2} (7 t^{3} - 13 t) + t^{2} \cdot 4}$

Step 7.2.5

Factor $t^{2}$ out of $t^{2} (7 t^{3} - 13 t) + t^{2} \cdot 4$ .

$\frac{17875}{t^{2} (7 t^{3} - 13 t + 4)}$