Calculus Examples

$\frac{lim_{x \to 5} 3 x^{2} - 13 x + 4}{2 x^{2} - 8 x}$

Step 1

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

$\frac{lim_{x \to 5} 3 x^{2} - lim_{x \to 5} 13 x + lim_{x \to 5} 4}{2 x^{2} - 8 x}$

Step 2

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 lim_{x \to 5} x^{2} - lim_{x \to 5} 13 x + lim_{x \to 5} 4}{2 x^{2} - 8 x}$

Step 3

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

$\frac{3 {(lim_{x \to 5} x)}^{2} - lim_{x \to 5} 13 x + lim_{x \to 5} 4}{2 x^{2} - 8 x}$

Step 4

Move the term $13$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 {(lim_{x \to 5} x)}^{2} - 13 lim_{x \to 5} x + lim_{x \to 5} 4}{2 x^{2} - 8 x}$

Step 5

Evaluate the limit of $4$ which is constant as $x$ approaches $5$ .

$\frac{3 {(lim_{x \to 5} x)}^{2} - 13 lim_{x \to 5} x + 4}{2 x^{2} - 8 x}$

Step 6

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

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Step 6.1

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

$\frac{3 \cdot 5^{2} - 13 lim_{x \to 5} x + 4}{2 x^{2} - 8 x}$

Step 6.2

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

$\frac{3 \cdot 5^{2} - 13 \cdot 5 + 4}{2 x^{2} - 8 x}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$\frac{3 \cdot 25 - 13 \cdot 5 + 4}{2 x^{2} - 8 x}$

Step 7.1.2

Multiply $3$ by $25$ .

$\frac{75 - 13 \cdot 5 + 4}{2 x^{2} - 8 x}$

Step 7.1.3

Multiply $- 13$ by $5$ .

$\frac{75 - 65 + 4}{2 x^{2} - 8 x}$

Step 7.1.4

Subtract $65$ from $75$ .

$\frac{10 + 4}{2 x^{2} - 8 x}$

Step 7.1.5

Add $10$ and $4$ .

$\frac{14}{2 x^{2} - 8 x}$

Step 7.2

Factor $2 x$ out of $2 x^{2} - 8 x$ .

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Step 7.2.1

Factor $2 x$ out of $2 x^{2}$ .

$\frac{14}{2 x (x) - 8 x}$

Step 7.2.2

Factor $2 x$ out of $- 8 x$ .

$\frac{14}{2 x (x) + 2 x (- 4)}$

Step 7.2.3

Factor $2 x$ out of $2 x (x) + 2 x (- 4)$ .

$\frac{14}{2 x (x - 4)}$

Step 7.3

Cancel the common factor of $14$ and $2$ .

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Step 7.3.1

Factor $2$ out of $14$ .

$\frac{2 \cdot 7}{2 x (x - 4)}$

Step 7.3.2

Cancel the common factors.

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Step 7.3.2.1

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

$\frac{2 \cdot 7}{2 (x (x - 4))}$

Step 7.3.2.2

Cancel the common factor.

$\frac{2 \cdot 7}{2 (x (x - 4))}$

Step 7.3.2.3

Rewrite the expression.

$\frac{7}{x (x - 4)}$