Calculus Examples

lim x \to 2 \frac{x^{2} + 3 x - 10}{x^{2} - 4}

$lim_{x \to 2} \frac{x^{2} + 3 x - 10}{x^{2} - 4}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

$\frac{lim_{x \to 2} x^{2} + 3 x - 10}{lim_{x \to 2} x^{2} - 4}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Split the limit using the Sum of Limits Rule on the limit as

$x$ approaches

$2$ .

$\frac{lim_{x \to 2} x^{2} + lim_{x \to 2} 3 x - lim_{x \to 2} 10}{lim_{x \to 2} x^{2} - 4}$

Step 1.1.2.2

Move the exponent

$2$ from

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

$\frac{{(lim_{x \to 2} x)}^{2} + lim_{x \to 2} 3 x - lim_{x \to 2} 10}{lim_{x \to 2} x^{2} - 4}$

Step 1.1.2.3

Move the term

$3$ outside of the limit because it is constant with respect to

$x$ .

$\frac{{(lim_{x \to 2} x)}^{2} + 3 lim_{x \to 2} x - lim_{x \to 2} 10}{lim_{x \to 2} x^{2} - 4}$

Step 1.1.2.4

Evaluate the limit of

$10$ which is constant as

$x$ approaches

$2$ .

$\frac{{(lim_{x \to 2} x)}^{2} + 3 lim_{x \to 2} x - 1 \cdot 10}{lim_{x \to 2} x^{2} - 4}$

Step 1.1.2.5

Evaluate the limits by plugging in

$2$ for all occurrences of

$x$ .

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Step 1.1.2.5.1

Evaluate the limit of

$x$ by plugging in

$2$ for

$x$ .

$\frac{2^{2} + 3 lim_{x \to 2} x - 1 \cdot 10}{lim_{x \to 2} x^{2} - 4}$

Step 1.1.2.5.2

Evaluate the limit of

$x$ by plugging in

$2$ for

$x$ .

$\frac{2^{2} + 3 \cdot 2 - 1 \cdot 10}{lim_{x \to 2} x^{2} - 4}$

Step 1.1.2.6

Simplify the answer.

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Step 1.1.2.6.1

Simplify each term.

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Step 1.1.2.6.1.1

Raise

$2$ to the power of

$2$ .

$\frac{4 + 3 \cdot 2 - 1 \cdot 10}{lim_{x \to 2} x^{2} - 4}$

Step 1.1.2.6.1.2

Multiply

$3$ by

$2$ .

$\frac{4 + 6 - 1 \cdot 10}{lim_{x \to 2} x^{2} - 4}$

Step 1.1.2.6.1.3

Multiply

$- 1$ by

$10$ .

$\frac{4 + 6 - 10}{lim_{x \to 2} x^{2} - 4}$

Step 1.1.2.6.2

Add

$4$ and

$6$ .

$\frac{10 - 10}{lim_{x \to 2} x^{2} - 4}$

Step 1.1.2.6.3

Subtract

$10$ from

$10$ .

$\frac{0}{lim_{x \to 2} x^{2} - 4}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as

$x$ approaches

$2$ .

$\frac{0}{lim_{x \to 2} x^{2} - lim_{x \to 2} 4}$

Step 1.1.3.1.2

Move the exponent

$2$ from

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

$\frac{0}{{(lim_{x \to 2} x)}^{2} - lim_{x \to 2} 4}$

Step 1.1.3.1.3

Evaluate the limit of

$4$ which is constant as

$x$ approaches

$2$ .

$\frac{0}{{(lim_{x \to 2} x)}^{2} - 1 \cdot 4}$

Step 1.1.3.2

Evaluate the limit of

$x$ by plugging in

$2$ for

$x$ .

$\frac{0}{2^{2} - 1 \cdot 4}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Raise

$2$ to the power of

$2$ .

$\frac{0}{4 - 1 \cdot 4}$

Step 1.1.3.3.1.2

Multiply

$- 1$ by

$4$ .

$\frac{0}{4 - 4}$

Step 1.1.3.3.2

Subtract

$4$ from

$4$ .

$\frac{0}{0}$

Step 1.1.3.3.3

The expression contains a division by

$0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.3.4

The expression contains a division by

$0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by

$0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.2

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.

$lim_{x \to 2} \frac{x^{2} + 3 x - 10}{x^{2} - 4} = lim_{x \to 2} \frac{\frac{d}{d x} [x^{2} + 3 x - 10]}{\frac{d}{d x} [x^{2} - 4]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 2} \frac{\frac{d}{d x} [x^{2} + 3 x - 10]}{\frac{d}{d x} [x^{2} - 4]}$

Step 1.3.2

By the Sum Rule, the derivative of

$x^{2} + 3 x - 10$ with respect to

$x$ is

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 10]$ .

$lim_{x \to 2} \frac{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 10]}{\frac{d}{d x} [x^{2} - 4]}$

Step 1.3.3

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 2$ .

$lim_{x \to 2} \frac{2 x + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 10]}{\frac{d}{d x} [x^{2} - 4]}$

Step 1.3.4

Evaluate

$\frac{d}{d x} [3 x]$ .

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Step 1.3.4.1

Since

$3$ is constant with respect to

$x$ , the derivative of

$3 x$ with respect to

$x$ is

$3 \frac{d}{d x} [x]$ .

$lim_{x \to 2} \frac{2 x + 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 10]}{\frac{d}{d x} [x^{2} - 4]}$

Step 1.3.4.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$lim_{x \to 2} \frac{2 x + 3 \cdot 1 + \frac{d}{d x} [- 10]}{\frac{d}{d x} [x^{2} - 4]}$

Step 1.3.4.3

Multiply

$3$ by

$1$ .

$lim_{x \to 2} \frac{2 x + 3 + \frac{d}{d x} [- 10]}{\frac{d}{d x} [x^{2} - 4]}$

Step 1.3.5

Since

$- 10$ is constant with respect to

$x$ , the derivative of

$- 10$ with respect to

$x$ is

$0$ .

$lim_{x \to 2} \frac{2 x + 3 + 0}{\frac{d}{d x} [x^{2} - 4]}$

Step 1.3.6

Add

$2 x + 3$ and

$0$ .

$lim_{x \to 2} \frac{2 x + 3}{\frac{d}{d x} [x^{2} - 4]}$

Step 1.3.7

By the Sum Rule, the derivative of

$x^{2} - 4$ with respect to

$x$ is

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]$ .

$lim_{x \to 2} \frac{2 x + 3}{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]}$

Step 1.3.8

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 2$ .

$lim_{x \to 2} \frac{2 x + 3}{2 x + \frac{d}{d x} [- 4]}$

Step 1.3.9

Since

$- 4$ is constant with respect to

$x$ , the derivative of

$- 4$ with respect to

$x$ is

$0$ .

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

Step 1.3.10

Add

$2 x$ and

$0$ .

$lim_{x \to 2} \frac{2 x + 3}{2 x}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term

$\frac{1}{2}$ outside of the limit because it is constant with respect to

$x$ .

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

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as

$x$ approaches

$2$ .

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

Step 2.3

Split the limit using the Sum of Limits Rule on the limit as

$x$ approaches

$2$ .

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

Step 2.4

Move the term

$2$ outside of the limit because it is constant with respect to

$x$ .

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

Step 2.5

Evaluate the limit of

$3$ which is constant as

$x$ approaches

$2$ .

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

Step 3

Evaluate the limits by plugging in

$2$ for all occurrences of

$x$ .

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Step 3.1

Evaluate the limit of

$x$ by plugging in

$2$ for

$x$ .

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

Step 3.2

Evaluate the limit of

$x$ by plugging in

$2$ for

$x$ .

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

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply

$2$ by

$2$ .

$\frac{1}{2} \cdot \frac{4 + 3}{2}$

Step 4.1.2

Add

$4$ and

$3$ .

$\frac{1}{2} \cdot \frac{7}{2}$

Step 4.2

Multiply

$\frac{1}{2} \cdot \frac{7}{2}$ .

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Step 4.2.1

Multiply

$\frac{1}{2}$ by

$\frac{7}{2}$ .

$\frac{7}{2 \cdot 2}$

Step 4.2.2

Multiply

$2$ by

$2$ .

$\frac{7}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{4}$

Decimal Form:

$1.75$