Calculus Examples

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

Step 1

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $0$ .

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

Step 2

Split the limit using the Product of Limits Rule on the limit as $t$ approaches $0$ .

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

Step 3

Move the limit under the radical sign.

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

Step 4

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $0$ .

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

Step 5

Evaluate the limit of $1$ which is constant as $t$ approaches $0$ .

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

Step 6

Evaluate the limits by plugging in $0$ for all occurrences of $t$ .

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Step 6.1

Evaluate the limit of $t$ by plugging in $0$ for $t$ .

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

Step 6.2

Evaluate the limit of $t$ by plugging in $0$ for $t$ .

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

Step 6.3

Evaluate the limit of $t$ by plugging in $0$ for $t$ .

$\frac{0 + 0 \cdot \sqrt{1 + 0}}{t^{2} \sqrt{1 + t}}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Add $1$ and $0$ .

$\frac{0 + 0 \cdot \sqrt{1}}{t^{2} \sqrt{1 + t}}$

Step 7.1.2

Any root of $1$ is $1$ .

$\frac{0 + 0 \cdot 1}{t^{2} \sqrt{1 + t}}$

Step 7.1.3

Multiply $0$ by $1$ .

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

Step 7.1.4

Add $0$ and $0$ .

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

Step 7.2

Divide $0$ by $t^{2} \sqrt{1 + t}$ .

$0$