Calculus Examples

$lim_{t \to 0} (\frac{9}{t \sqrt{1 + t}}) (\frac{9}{t})$

Step 1

Combine factors.

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Step 1.1

Multiply $\frac{9}{t \sqrt{1 + t}}$ by $\frac{9}{t}$ .

$lim_{t \to 0} \frac{9 \cdot 9}{t \sqrt{1 + t} t}$

Step 1.2

Multiply $9$ by $9$ .

$lim_{t \to 0} \frac{81}{t \sqrt{1 + t} t}$

Step 1.3

Raise $t$ to the power of $1$ .

$lim_{t \to 0} \frac{81}{t^{1} t \sqrt{1 + t}}$

Step 1.4

Raise $t$ to the power of $1$ .

$lim_{t \to 0} \frac{81}{t^{1} t^{1} \sqrt{1 + t}}$

Step 1.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{t \to 0} \frac{81}{t^{1 + 1} \sqrt{1 + t}}$

Step 1.6

Add $1$ and $1$ .

$lim_{t \to 0} \frac{81}{t^{2} \sqrt{1 + t}}$

Step 2

Since the numerator is positive and the denominator $t^{2} \sqrt{1 + t}$ approaches zero and is greater than zero for $t$ near $0$ on both sides, the function increases without bound.

$\infty$