Calculus Examples

$lim_{t \to \infty} 24 - \frac{16 t^{2}}{{(t + 2)}^{2}}$

Step 1

Evaluate the limit.

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Step 1.1

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

$lim_{t \to \infty} 24 - lim_{t \to \infty} \frac{16 t^{2}}{{(t + 2)}^{2}}$

Step 1.2

Evaluate the limit of $24$ which is constant as $t$ approaches $\infty$ .

$24 - lim_{t \to \infty} \frac{16 t^{2}}{{(t + 2)}^{2}}$

Step 1.3

Move the term $16$ outside of the limit because it is constant with respect to $t$ .

$24 - 16 lim_{t \to \infty} \frac{t^{2}}{{(t + 2)}^{2}}$

Step 2

Divide the numerator and denominator by the highest power of $t$ in the denominator.

$24 - 16 lim_{t \to \infty} \frac{\frac{t^{2}}{t^{2}}}{{(\frac{t}{t} + \frac{2}{t})}^{2}}$

Step 3

Evaluate the limit.

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Step 3.1

Cancel the common factor of $t^{2}$ .

$24 - 16 lim_{t \to \infty} \frac{1}{{(\frac{t}{t} + \frac{2}{t})}^{2}}$

Step 3.2

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

$24 - 16 \frac{lim_{t \to \infty} 1}{lim_{t \to \infty} {(\frac{t}{t} + \frac{2}{t})}^{2}}$

Step 3.3

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

$24 - 16 \frac{1}{lim_{t \to \infty} {(\frac{t}{t} + \frac{2}{t})}^{2}}$

Step 3.4

Move the exponent $2$ from ${(\frac{t}{t} + \frac{2}{t})}^{2}$ outside the limit using the Limits Power Rule.

$24 - 16 \frac{1}{{(lim_{t \to \infty} \frac{t}{t} + \frac{2}{t})}^{2}}$

Step 3.5

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

$24 - 16 \frac{1}{{(lim_{t \to \infty} \frac{t}{t} + lim_{t \to \infty} \frac{2}{t})}^{2}}$

Step 3.6

Cancel the common factor of $t$ .

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Step 3.6.1

Cancel the common factor.

$24 - 16 \frac{1}{{(lim_{t \to \infty} \frac{t}{t} + lim_{t \to \infty} \frac{2}{t})}^{2}}$

Step 3.6.2

Rewrite the expression.

$24 - 16 \frac{1}{{(lim_{t \to \infty} 1 + lim_{t \to \infty} \frac{2}{t})}^{2}}$

Step 3.7

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

$24 - 16 \frac{1}{{(1 + lim_{t \to \infty} \frac{2}{t})}^{2}}$

Step 3.8

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

$24 - 16 \frac{1}{{(1 + 2 lim_{t \to \infty} \frac{1}{t})}^{2}}$

Step 4

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{t}$ approaches $0$ .

$24 - 16 \frac{1}{{(1 + 2 \cdot 0)}^{2}}$

Step 5

Simplify the answer.

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Step 5.1

Simplify each term.

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Step 5.1.1

Simplify the denominator.

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Step 5.1.1.1

Multiply $2$ by $0$ .

$24 - 16 \frac{1}{{(1 + 0)}^{2}}$

Step 5.1.1.2

Add $1$ and $0$ .

$24 - 16 \frac{1}{1^{2}}$

Step 5.1.1.3

One to any power is one.

$24 - 16 (\frac{1}{1})$

Step 5.1.2

Cancel the common factor of $1$ .

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Step 5.1.2.1

Cancel the common factor.

$24 - 16 (\frac{1}{1})$

Step 5.1.2.2

Rewrite the expression.

$24 - 16 \cdot 1$

Step 5.1.3

Multiply $- 16$ by $1$ .

$24 - 16$

Step 5.2

Subtract $16$ from $24$ .

$8$