Calculus Examples

$\frac{lim_{t \to 15} t^{2} + 3 t - 270}{t^{2} - 225}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $15$ .

$\frac{lim_{t \to 15} t^{2} + lim_{t \to 15} 3 t - lim_{t \to 15} 270}{t^{2} - 225}$

Step 2

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

$\frac{{(lim_{t \to 15} t)}^{2} + lim_{t \to 15} 3 t - lim_{t \to 15} 270}{t^{2} - 225}$

Step 3

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

$\frac{{(lim_{t \to 15} t)}^{2} + 3 lim_{t \to 15} t - lim_{t \to 15} 270}{t^{2} - 225}$

Step 4

Evaluate the limit of $270$ which is constant as $t$ approaches $15$ .

$\frac{{(lim_{t \to 15} t)}^{2} + 3 lim_{t \to 15} t - 1 \cdot 270}{t^{2} - 225}$

Step 5

Evaluate the limits by plugging in $15$ for all occurrences of $t$ .

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Step 5.1

Evaluate the limit of $t$ by plugging in $15$ for $t$ .

$\frac{15^{2} + 3 lim_{t \to 15} t - 1 \cdot 270}{t^{2} - 225}$

Step 5.2

Evaluate the limit of $t$ by plugging in $15$ for $t$ .

$\frac{15^{2} + 3 \cdot 15 - 1 \cdot 270}{t^{2} - 225}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $15$ to the power of $2$ .

$\frac{225 + 3 \cdot 15 - 1 \cdot 270}{t^{2} - 225}$

Step 6.1.2

Multiply $3$ by $15$ .

$\frac{225 + 45 - 1 \cdot 270}{t^{2} - 225}$

Step 6.1.3

Multiply $- 1$ by $270$ .

$\frac{225 + 45 - 270}{t^{2} - 225}$

Step 6.1.4

Add $225$ and $45$ .

$\frac{270 - 270}{t^{2} - 225}$

Step 6.1.5

Subtract $270$ from $270$ .

$\frac{0}{t^{2} - 225}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Rewrite $225$ as $15^{2}$ .

$\frac{0}{t^{2} - 15^{2}}$

Step 6.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = t$ and $b = 15$ .

$\frac{0}{(t + 15) (t - 15)}$

Step 6.3

Divide $0$ by $(t + 15) (t - 15)$ .

$0$