Simplify limit as x approaches 5 of (x^2-2x-15)/(x^2-10x+25)

$lim_{x \to 5} \frac{x^{2} - 2 x - 15}{x^{2} - 10 x + 25}$

Step 1

Apply L\'Hospital\'s rule.

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

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

$\frac{lim_{x \to 5} x^{2} - 2 x - 15}{lim_{x \to 5} x^{2} - 10 x + 25}$

Evaluate the limit of the numerator.

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

$\frac{lim_{x \to 5} x^{2} - lim_{x \to 5} 2 x - lim_{x \to 5} 15}{lim_{x \to 5} x^{2} - 10 x + 25}$

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

$\frac{{(lim_{x \to 5} x)}^{2} - lim_{x \to 5} 2 x - lim_{x \to 5} 15}{lim_{x \to 5} x^{2} - 10 x + 25}$

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 5} x)}^{2} - 2 lim_{x \to 5} x - lim_{x \to 5} 15}{lim_{x \to 5} x^{2} - 10 x + 25}$

Evaluate the limit of $15$ which is constant as $x$ approaches $5$ .

$\frac{{(lim_{x \to 5} x)}^{2} - 2 lim_{x \to 5} x - 1 \cdot 15}{lim_{x \to 5} x^{2} - 10 x + 25}$

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

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

$\frac{5^{2} - 2 lim_{x \to 5} x - 1 \cdot 15}{lim_{x \to 5} x^{2} - 10 x + 25}$

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

$\frac{5^{2} - 2 \cdot 5 - 1 \cdot 15}{lim_{x \to 5} x^{2} - 10 x + 25}$

Simplify the answer.

Simplify each term.

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

$\frac{25 - 2 \cdot 5 - 1 \cdot 15}{lim_{x \to 5} x^{2} - 10 x + 25}$

Multiply $- 2$ by $5$ .

$\frac{25 - 10 - 1 \cdot 15}{lim_{x \to 5} x^{2} - 10 x + 25}$

Multiply $- 1$ by $15$ .

$\frac{25 - 10 - 15}{lim_{x \to 5} x^{2} - 10 x + 25}$

Subtract $10$ from $25$ .

$\frac{15 - 15}{lim_{x \to 5} x^{2} - 10 x + 25}$

Subtract $15$ from $15$ .

$\frac{0}{lim_{x \to 5} x^{2} - 10 x + 25}$

Evaluate the limit of the denominator.

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

$\frac{0}{lim_{x \to 5} x^{2} - lim_{x \to 5} 10 x + lim_{x \to 5} 25}$

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

$\frac{0}{{(lim_{x \to 5} x)}^{2} - lim_{x \to 5} 10 x + lim_{x \to 5} 25}$

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

$\frac{0}{{(lim_{x \to 5} x)}^{2} - 10 lim_{x \to 5} x + lim_{x \to 5} 25}$

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

$\frac{0}{{(lim_{x \to 5} x)}^{2} - 10 lim_{x \to 5} x + 25}$

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

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

$\frac{0}{5^{2} - 10 lim_{x \to 5} x + 25}$

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

$\frac{0}{5^{2} - 10 \cdot 5 + 25}$

Simplify the answer.

Simplify each term.

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

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

Multiply $- 10$ by $5$ .

$\frac{0}{25 - 50 + 25}$

Subtract $50$ from $25$ .

$\frac{0}{- 25 + 25}$

Add $- 25$ and $25$ .

$\frac{0}{0}$

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

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.

$lim_{x \to 5} \frac{x^{2} - 2 x - 15}{x^{2} - 10 x + 25} = lim_{x \to 5} \frac{\frac{d}{d x} [x^{2} - 2 x - 15]}{\frac{d}{d x} [x^{2} - 10 x + 25]}$

Find the derivative of the numerator and denominator.

Differentiate the numerator and denominator.

$lim_{x \to 5} \frac{\frac{d}{d x} [x^{2} - 2 x - 15]}{\frac{d}{d x} [x^{2} - 10 x + 25]}$

By the Sum Rule, the derivative of $x^{2} - 2 x - 15$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 15]$ .

$lim_{x \to 5} \frac{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 15]}{\frac{d}{d x} [x^{2} - 10 x + 25]}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 5} \frac{2 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 15]}{\frac{d}{d x} [x^{2} - 10 x + 25]}$

Evaluate $\frac{d}{d x} [- 2 x]$ .

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$lim_{x \to 5} \frac{2 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 15]}{\frac{d}{d x} [x^{2} - 10 x + 25]}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 5} \frac{2 x - 2 \cdot 1 + \frac{d}{d x} [- 15]}{\frac{d}{d x} [x^{2} - 10 x + 25]}$

Multiply $- 2$ by $1$ .

$lim_{x \to 5} \frac{2 x - 2 + \frac{d}{d x} [- 15]}{\frac{d}{d x} [x^{2} - 10 x + 25]}$

Since $- 15$ is constant with respect to $x$ , the derivative of $- 15$ with respect to $x$ is $0$ .

$lim_{x \to 5} \frac{2 x - 2 + 0}{\frac{d}{d x} [x^{2} - 10 x + 25]}$

Add $2 x - 2$ and $0$ .

$lim_{x \to 5} \frac{2 x - 2}{\frac{d}{d x} [x^{2} - 10 x + 25]}$

By the Sum Rule, the derivative of $x^{2} - 10 x + 25$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [25]$ .

$lim_{x \to 5} \frac{2 x - 2}{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [25]}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 5} \frac{2 x - 2}{2 x + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [25]}$

Evaluate $\frac{d}{d x} [- 10 x]$ .

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 x$ with respect to $x$ is $- 10 \frac{d}{d x} [x]$ .

$lim_{x \to 5} \frac{2 x - 2}{2 x - 10 \frac{d}{d x} [x] + \frac{d}{d x} [25]}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 5} \frac{2 x - 2}{2 x - 10 \cdot 1 + \frac{d}{d x} [25]}$

Multiply $- 10$ by $1$ .

$lim_{x \to 5} \frac{2 x - 2}{2 x - 10 + \frac{d}{d x} [25]}$

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$lim_{x \to 5} \frac{2 x - 2}{2 x - 10 + 0}$

Add $2 x - 10$ and $0$ .

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

Cancel the common factor of $2 x - 2$ and $2 x - 10$ .

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $- 2$ .

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

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

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

Cancel the common factors.

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $- 10$ .

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

Factor $2$ out of $2 x + 2 \cdot - 5$ .

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

Cancel the common factor.

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

Rewrite the expression.

$lim_{x \to 5} \frac{x - 1}{x - 5}$

Step 2

Since the function approaches $- \infty$ from the left and $\infty$ from the right, the limit does not exist.

$Does not exist$

Simplify limit as x approaches 5 of (x^2-2x-15)/(x^2-10x+25)

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