Calculus Examples

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

Step 1

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

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

Step 2

Move the limit under the radical sign.

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

Step 3

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

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

Step 4

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

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

Step 5

Move the limit under the radical sign.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

$\frac{\sqrt{1 + 0} - \sqrt{1 + 0}}{t}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Add $1$ and $0$ .

$\frac{\sqrt{1} - \sqrt{1 + 0}}{t}$

Step 9.1.2

Any root of $1$ is $1$ .

$\frac{1 - \sqrt{1 + 0}}{t}$

Step 9.1.3

Add $1$ and $0$ .

$\frac{1 - \sqrt{1}}{t}$

Step 9.1.4

Any root of $1$ is $1$ .

$\frac{1 - 1 \cdot 1}{t}$

Step 9.1.5

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{t}$

Step 9.1.6

Subtract $1$ from $1$ .

$\frac{0}{t}$

Step 9.2

Divide $0$ by $t$ .

$0$