Calculus Examples

$lim_{x \to 5} \frac{x^{2} - 3 x - 10}{5 x^{2} - 10 x + 25}$

Step 1

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

$\frac{lim_{x \to 5} x^{2} - 3 x - 10}{lim_{x \to 5} 5 x^{2} - 10 x + 25}$

Step 2

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

$\frac{lim_{x \to 5} x^{2} - lim_{x \to 5} 3 x - lim_{x \to 5} 10}{lim_{x \to 5} 5 x^{2} - 10 x + 25}$

Step 3

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

$\frac{{(lim_{x \to 5} x)}^{2} - lim_{x \to 5} 3 x - lim_{x \to 5} 10}{lim_{x \to 5} 5 x^{2} - 10 x + 25}$

Step 4

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

$\frac{{(lim_{x \to 5} x)}^{2} - 3 lim_{x \to 5} x - lim_{x \to 5} 10}{lim_{x \to 5} 5 x^{2} - 10 x + 25}$

Step 5

Evaluate the limit of $10$ which is constant as $x$ approaches $5$ .

$\frac{{(lim_{x \to 5} x)}^{2} - 3 lim_{x \to 5} x - 1 \cdot 10}{lim_{x \to 5} 5 x^{2} - 10 x + 25}$

Step 6

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

$\frac{{(lim_{x \to 5} x)}^{2} - 3 lim_{x \to 5} x - 1 \cdot 10}{lim_{x \to 5} 5 x^{2} - lim_{x \to 5} 10 x + lim_{x \to 5} 25}$

Step 7

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 5} x)}^{2} - 3 lim_{x \to 5} x - 1 \cdot 10}{5 lim_{x \to 5} x^{2} - lim_{x \to 5} 10 x + lim_{x \to 5} 25}$

Step 8

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

$\frac{{(lim_{x \to 5} x)}^{2} - 3 lim_{x \to 5} x - 1 \cdot 10}{5 {(lim_{x \to 5} x)}^{2} - lim_{x \to 5} 10 x + lim_{x \to 5} 25}$

Step 9

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

$\frac{{(lim_{x \to 5} x)}^{2} - 3 lim_{x \to 5} x - 1 \cdot 10}{5 {(lim_{x \to 5} x)}^{2} - 10 lim_{x \to 5} x + lim_{x \to 5} 25}$

Step 10

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

$\frac{{(lim_{x \to 5} x)}^{2} - 3 lim_{x \to 5} x - 1 \cdot 10}{5 {(lim_{x \to 5} x)}^{2} - 10 lim_{x \to 5} x + 25}$

Step 11

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

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Step 11.1

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

$\frac{5^{2} - 3 lim_{x \to 5} x - 1 \cdot 10}{5 {(lim_{x \to 5} x)}^{2} - 10 lim_{x \to 5} x + 25}$

Step 11.2

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

$\frac{5^{2} - 3 \cdot 5 - 1 \cdot 10}{5 {(lim_{x \to 5} x)}^{2} - 10 lim_{x \to 5} x + 25}$

Step 11.3

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

$\frac{5^{2} - 3 \cdot 5 - 1 \cdot 10}{5 \cdot 5^{2} - 10 lim_{x \to 5} x + 25}$

Step 11.4

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

$\frac{5^{2} - 3 \cdot 5 - 1 \cdot 10}{5 \cdot 5^{2} - 10 \cdot 5 + 25}$

Step 12

Simplify the answer.

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Step 12.1

Cancel the common factor of $5^{2} - 3 \cdot 5 - 1 \cdot 10$ and $5 \cdot 5^{2} - 10 \cdot 5 + 25$ .

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Step 12.1.1

Factor $- 1$ out of $5^{2}$ .

$\frac{- 1 (- 5^{2}) - 3 \cdot 5 - 1 \cdot 10}{5 \cdot 5^{2} - 10 \cdot 5 + 25}$

Step 12.1.2

Factor $- 1$ out of $- 3 \cdot 5$ .

$\frac{- 1 (- 5^{2}) - (3 \cdot 5) - 1 \cdot 10}{5 \cdot 5^{2} - 10 \cdot 5 + 25}$

Step 12.1.3

Factor $- 1$ out of $- 1 (- 5^{2}) - (3 \cdot 5)$ .

$\frac{- 1 (- 5^{2} + 3 \cdot 5) - 1 \cdot 10}{5 \cdot 5^{2} - 10 \cdot 5 + 25}$

Step 12.1.4

Factor $- 1$ out of $- 1 \cdot 10$ .

$\frac{- 1 (- 5^{2} + 3 \cdot 5) - 1 (10)}{5 \cdot 5^{2} - 10 \cdot 5 + 25}$

Step 12.1.5

Factor $- 1$ out of $- 1 (- 5^{2} + 3 \cdot 5) - 1 (10)$ .

$\frac{- 1 (- 5^{2} + 3 \cdot 5 + 10)}{5 \cdot 5^{2} - 10 \cdot 5 + 25}$

Step 12.1.6

Factor $5$ out of $- 1 (- 5^{2} + 3 \cdot 5 + 10)$ .

$\frac{5 (- 1 (- 1 \cdot 5 + 3 + 2))}{5 \cdot 5^{2} - 10 \cdot 5 + 25}$

Step 12.1.7

Cancel the common factors.

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Step 12.1.7.1

Factor $5$ out of $5 \cdot 5^{2}$ .

$\frac{5 (- 1 (- 1 \cdot 5 + 3 + 2))}{5 (5^{2}) - 10 \cdot 5 + 25}$

Step 12.1.7.2

Factor $5$ out of $- 10 \cdot 5$ .

$\frac{5 (- 1 (- 1 \cdot 5 + 3 + 2))}{5 (5^{2}) + 5 (- 2 \cdot 5) + 25}$

Step 12.1.7.3

Factor $5$ out of $5 (5^{2}) + 5 (- 2 \cdot 5)$ .

$\frac{5 (- 1 (- 1 \cdot 5 + 3 + 2))}{5 (5^{2} - 2 \cdot 5) + 25}$

Step 12.1.7.4

Factor $5$ out of $25$ .

$\frac{5 (- 1 (- 1 \cdot 5 + 3 + 2))}{5 (5^{2} - 2 \cdot 5) + 5 (5)}$

Step 12.1.7.5

Factor $5$ out of $5 (5^{2} - 2 \cdot 5) + 5 (5)$ .

$\frac{5 (- 1 (- 1 \cdot 5 + 3 + 2))}{5 (5^{2} - 2 \cdot 5 + 5)}$

Step 12.1.7.6

Cancel the common factor.

$\frac{5 (- 1 (- 1 \cdot 5 + 3 + 2))}{5 (5^{2} - 2 \cdot 5 + 5)}$

Step 12.1.7.7

Rewrite the expression.

$\frac{- 1 (- 1 \cdot 5 + 3 + 2)}{5^{2} - 2 \cdot 5 + 5}$

Step 12.2

Simplify the numerator.

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Step 12.2.1

Multiply $- 1$ by $5$ .

$\frac{- 1 (- 5 + 3 + 2)}{5^{2} - 2 \cdot 5 + 5}$

Step 12.2.2

Add $- 5$ and $3$ .

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

Step 12.2.3

Add $- 2$ and $2$ .

$\frac{- 1 \cdot 0}{5^{2} - 2 \cdot 5 + 5}$

Step 12.3

Simplify the denominator.

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Step 12.3.1

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

$\frac{- 1 \cdot 0}{25 - 2 \cdot 5 + 5}$

Step 12.3.2

Multiply $- 2$ by $5$ .

$\frac{- 1 \cdot 0}{25 - 10 + 5}$

Step 12.3.3

Subtract $10$ from $25$ .

$\frac{- 1 \cdot 0}{15 + 5}$

Step 12.3.4

Add $15$ and $5$ .

$\frac{- 1 \cdot 0}{20}$

Step 12.4

Multiply $- 1$ by $0$ .

$\frac{0}{20}$

Step 12.5

Divide $0$ by $20$ .

$0$