Calculus Examples

$lim_{x \to 5} \frac{x^{2} - 25}{x - 5} + \sqrt{x^{2} + 7}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$lim_{x \to 5} \frac{x^{2} - 25}{x - 5} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 2.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 5} x^{2} - 25}{lim_{x \to 5} x - 5} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Evaluate the limit.

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Step 2.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{lim_{x \to 5} x^{2} - lim_{x \to 5} 25}{lim_{x \to 5} x - 5} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.1.2.1.2

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 5} x)}^{2} - lim_{x \to 5} 25}{lim_{x \to 5} x - 5} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.1.2.1.3

Evaluate the limit of $25$ which is constant as $x$ approaches $5$ .

$\frac{{(lim_{x \to 5} x)}^{2} - 1 \cdot 25}{lim_{x \to 5} x - 5} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.1.2.2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{5^{2} - 1 \cdot 25}{lim_{x \to 5} x - 5} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

Raise $5$ to the power of $2$ .

$\frac{25 - 1 \cdot 25}{lim_{x \to 5} x - 5} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.1.2.3.1.2

Multiply $- 1$ by $25$ .

$\frac{25 - 25}{lim_{x \to 5} x - 5} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.1.2.3.2

Subtract $25$ from $25$ .

$\frac{0}{lim_{x \to 5} x - 5} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Evaluate the limit.

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Step 2.1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{0}{lim_{x \to 5} x - lim_{x \to 5} 5} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.1.3.1.2

Evaluate the limit of $5$ which is constant as $x$ approaches $5$ .

$\frac{0}{lim_{x \to 5} x - 1 \cdot 5} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.1.3.2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{0}{5 - 1 \cdot 5} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.1.3.3

Simplify the answer.

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Step 2.1.3.3.1

Multiply $- 1$ by $5$ .

$\frac{0}{5 - 5} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.1.3.3.2

Subtract $5$ from $5$ .

$\frac{0}{0} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.1.3.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.1.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 5} \frac{x^{2} - 25}{x - 5} = lim_{x \to 5} \frac{\frac{d}{d x} [x^{2} - 25]}{\frac{d}{d x} [x - 5]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{x \to 5} \frac{\frac{d}{d x} [x^{2} - 25]}{\frac{d}{d x} [x - 5]} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.3.2

By the Sum Rule, the derivative of $x^{2} - 25$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 25]$ .

$lim_{x \to 5} \frac{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 25]}{\frac{d}{d x} [x - 5]} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 5} \frac{2 x + \frac{d}{d x} [- 25]}{\frac{d}{d x} [x - 5]} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.3.4

Since $- 25$ is constant with respect to $x$ , the derivative of $- 25$ with respect to $x$ is $0$ .

$lim_{x \to 5} \frac{2 x + 0}{\frac{d}{d x} [x - 5]} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.3.5

Add $2 x$ and $0$ .

$lim_{x \to 5} \frac{2 x}{\frac{d}{d x} [x - 5]} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.3.6

By the Sum Rule, the derivative of $x - 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

$lim_{x \to 5} \frac{2 x}{\frac{d}{d x} [x] + \frac{d}{d x} [- 5]} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.3.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 5} \frac{2 x}{1 + \frac{d}{d x} [- 5]} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.3.8

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$lim_{x \to 5} \frac{2 x}{1 + 0} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.3.9

Add $1$ and $0$ .

$lim_{x \to 5} \frac{2 x}{1} + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 2.4

Divide $2 x$ by $1$ .

$lim_{x \to 5} 2 x + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 3

Evaluate the limit.

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Step 3.1

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

$2 lim_{x \to 5} x + lim_{x \to 5} \sqrt{x^{2} + 7}$

Step 3.2

Move the limit under the radical sign.

$2 lim_{x \to 5} x + \sqrt{lim_{x \to 5} x^{2} + 7}$

Step 3.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$2 lim_{x \to 5} x + \sqrt{lim_{x \to 5} x^{2} + lim_{x \to 5} 7}$

Step 3.4

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$2 lim_{x \to 5} x + \sqrt{{(lim_{x \to 5} x)}^{2} + lim_{x \to 5} 7}$

Step 3.5

Evaluate the limit of $7$ which is constant as $x$ approaches $5$ .

$2 lim_{x \to 5} x + \sqrt{{(lim_{x \to 5} x)}^{2} + 7}$

Step 4

Evaluate the limits by plugging in $5$ for all occurrences of $x$ .

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Step 4.1

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$2 \cdot 5 + \sqrt{{(lim_{x \to 5} x)}^{2} + 7}$

Step 4.2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$2 \cdot 5 + \sqrt{5^{2} + 7}$

Step 5

Simplify each term.

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Step 5.1

Multiply $2$ by $5$ .

$10 + \sqrt{5^{2} + 7}$

Step 5.2

Raise $5$ to the power of $2$ .

$10 + \sqrt{25 + 7}$

Step 5.3

Add $25$ and $7$ .

$10 + \sqrt{32}$

Step 5.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 5.4.1

Factor $16$ out of $32$ .

$10 + \sqrt{16 (2)}$

Step 5.4.2

Rewrite $16$ as $4^{2}$ .

$10 + \sqrt{4^{2} \cdot 2}$

Step 5.5

Pull terms out from under the radical.

$10 + 4 \sqrt{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$10 + 4 \sqrt{2}$

Decimal Form:

$15.65685424 \dots$