Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

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

Step 1.2

Move the exponent $2$ from ${(10 x - 5)}^{2}$ outside the limit using the Limits Power Rule.

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

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

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

Step 1.4

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

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

Step 1.5

Evaluate the limit of $5$ which is constant as $x$ approaches $1$ .

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

Step 1.6

Evaluate the limit of $25$ which is constant as $x$ approaches $1$ .

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

Step 2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Multiply $10$ by $1$ .

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

Step 3.1.1.2

Multiply $- 1$ by $5$ .

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

Step 3.1.2

Subtract $5$ from $10$ .

$\frac{5^{2} - 1 \cdot 25}{2 x - 2}$

Step 3.1.3

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

$\frac{25 - 1 \cdot 25}{2 x - 2}$

Step 3.1.4

Multiply $- 1$ by $25$ .

$\frac{25 - 25}{2 x - 2}$

Step 3.1.5

Subtract $25$ from $25$ .

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

Step 3.2

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

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Step 3.2.1

Factor $2$ out of $2 x$ .

$\frac{0}{2 (x) - 2}$

Step 3.2.2

Factor $2$ out of $- 2$ .

$\frac{0}{2 (x) + 2 (- 1)}$

Step 3.2.3

Factor $2$ out of $2 (x) + 2 (- 1)$ .

$\frac{0}{2 (x - 1)}$

Step 3.3

Cancel the common factor of $0$ and $2$ .

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Step 3.3.1

Factor $2$ out of $0$ .

$\frac{2 (0)}{2 (x - 1)}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Cancel the common factor.

$\frac{2 \cdot 0}{2 (x - 1)}$

Step 3.3.2.2

Rewrite the expression.

$\frac{0}{x - 1}$

Step 3.4

Divide $0$ by $x - 1$ .

$0$