Calculus Examples

$lim_{x \to 4} \frac{10}{\frac{x}{x - 4}} - \frac{5}{2}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$lim_{x \to 4} \frac{10}{\frac{x}{x - 4}} - lim_{x \to 4} \frac{5}{2}$

Step 2

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$10 lim_{x \to 4} \frac{1}{\frac{x}{x - 4}} - lim_{x \to 4} \frac{5}{2}$

Step 3

Simplify the limit argument.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$10 lim_{x \to 4} 1 \frac{x - 4}{x} - lim_{x \to 4} \frac{5}{2}$

Step 3.2

Multiply $\frac{x - 4}{x}$ by $1$ .

$10 lim_{x \to 4} \frac{x - 4}{x} - lim_{x \to 4} \frac{5}{2}$

Step 4

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4$ .

$10 \frac{lim_{x \to 4} x - 4}{lim_{x \to 4} x} - lim_{x \to 4} \frac{5}{2}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$10 \frac{lim_{x \to 4} x - lim_{x \to 4} 4}{lim_{x \to 4} x} - lim_{x \to 4} \frac{5}{2}$

Step 6

Evaluate the limit of $4$ which is constant as $x$ approaches $4$ .

$10 \frac{lim_{x \to 4} x - 1 \cdot 4}{lim_{x \to 4} x} - lim_{x \to 4} \frac{5}{2}$

Step 7

Evaluate the limit of $\frac{5}{2}$ which is constant as $x$ approaches $4$ .

$10 \frac{lim_{x \to 4} x - 1 \cdot 4}{lim_{x \to 4} x} - \frac{5}{2}$

Step 8

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$10 \frac{4 - 1 \cdot 4}{lim_{x \to 4} x} - \frac{5}{2}$

Step 8.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$10 \frac{4 - 1 \cdot 4}{4} - \frac{5}{2}$

Step 9

Simplify the answer.

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Step 9.1

Simplify each term.

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Step 9.1.1

Cancel the common factor of $2$ .

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Step 9.1.1.1

Factor $2$ out of $10$ .

$2 (5) \frac{4 - 1 \cdot 4}{4} - \frac{5}{2}$

Step 9.1.1.2

Factor $2$ out of $4$ .

$2 \cdot 5 \frac{4 - 1 \cdot 4}{2 \cdot 2} - \frac{5}{2}$

Step 9.1.1.3

Cancel the common factor.

$2 \cdot 5 \frac{4 - 1 \cdot 4}{2 \cdot 2} - \frac{5}{2}$

Step 9.1.1.4

Rewrite the expression.

$5 \frac{4 - 1 \cdot 4}{2} - \frac{5}{2}$

Step 9.1.2

Multiply $- 1$ by $4$ .

$5 \frac{4 - 4}{2} - \frac{5}{2}$

Step 9.1.3

Combine $5$ and $\frac{4 - 4}{2}$ .

$\frac{5 (4 - 4)}{2} - \frac{5}{2}$

Step 9.1.4

Cancel the common factor of $4 - 4$ and $2$ .

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Step 9.1.4.1

Factor $2$ out of $5 (4 - 4)$ .

$\frac{2 (5 (2 - 2))}{2} - \frac{5}{2}$

Step 9.1.4.2

Cancel the common factors.

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Step 9.1.4.2.1

Factor $2$ out of $2$ .

$\frac{2 (5 (2 - 2))}{2 (1)} - \frac{5}{2}$

Step 9.1.4.2.2

Cancel the common factor.

$\frac{2 (5 (2 - 2))}{2 \cdot 1} - \frac{5}{2}$

Step 9.1.4.2.3

Rewrite the expression.

$\frac{5 (2 - 2)}{1} - \frac{5}{2}$

Step 9.1.4.2.4

Divide $5 (2 - 2)$ by $1$ .

$5 (2 - 2) - \frac{5}{2}$

Step 9.1.5

Subtract $2$ from $2$ .

$5 \cdot 0 - \frac{5}{2}$

Step 9.1.6

Multiply $5$ by $0$ .

$0 - \frac{5}{2}$

Step 9.2

Subtract $\frac{5}{2}$ from $0$ .

$- \frac{5}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{5}{2}$

Decimal Form:

$- 2.5$