Calculus Examples

$lim_{x \to 0} \frac{x - 3}{\sqrt{x + 1} - 2}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} x - 3}{lim_{x \to 0} \sqrt{x + 1} - 2}$

Step 2

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

$\frac{lim_{x \to 0} x - lim_{x \to 0} 3}{lim_{x \to 0} \sqrt{x + 1} - 2}$

Step 3

Evaluate the limit of $3$ which is constant as $x$ approaches $0$ .

$\frac{lim_{x \to 0} x - 1 \cdot 3}{lim_{x \to 0} \sqrt{x + 1} - 2}$

Step 4

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

$\frac{lim_{x \to 0} x - 1 \cdot 3}{lim_{x \to 0} \sqrt{x + 1} - lim_{x \to 0} 2}$

Step 5

Move the limit under the radical sign.

$\frac{lim_{x \to 0} x - 1 \cdot 3}{\sqrt{lim_{x \to 0} x + 1} - lim_{x \to 0} 2}$

Step 6

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

$\frac{lim_{x \to 0} x - 1 \cdot 3}{\sqrt{lim_{x \to 0} x + lim_{x \to 0} 1} - lim_{x \to 0} 2}$

Step 7

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

$\frac{lim_{x \to 0} x - 1 \cdot 3}{\sqrt{lim_{x \to 0} x + 1} - lim_{x \to 0} 2}$

Step 8

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$\frac{lim_{x \to 0} x - 1 \cdot 3}{\sqrt{lim_{x \to 0} x + 1} - 1 \cdot 2}$

Step 9

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

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Step 9.1

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

$\frac{0 - 1 \cdot 3}{\sqrt{lim_{x \to 0} x + 1} - 1 \cdot 2}$

Step 9.2

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

$\frac{0 - 1 \cdot 3}{\sqrt{0 + 1} - 1 \cdot 2}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Multiply $- 1$ by $3$ .

$\frac{0 - 3}{\sqrt{0 + 1} - 1 \cdot 2}$

Step 10.1.2

Subtract $3$ from $0$ .

$\frac{- 3}{\sqrt{0 + 1} - 1 \cdot 2}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Add $0$ and $1$ .

$\frac{- 3}{\sqrt{1} - 1 \cdot 2}$

Step 10.2.2

Any root of $1$ is $1$ .

$\frac{- 3}{1 - 1 \cdot 2}$

Step 10.2.3

Multiply $- 1$ by $2$ .

$\frac{- 3}{1 - 2}$

Step 10.2.4

Subtract $2$ from $1$ .

$\frac{- 3}{- 1}$

Step 10.3

Divide $- 3$ by $- 1$ .

$3$