Simplify limit as x approaches 6 of (x^2-3x-18)/(x^2-12x+36)

$lim_{x \to 6} \frac{x^{2} - 3 x - 18}{x^{2} - 12 x + 36}$

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.

$\frac{lim_{x \to 6} x^{2} - 3 x - 18}{lim_{x \to 6} x^{2} - 12 x + 36}$

Evaluate the limit of the numerator.

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $6$ .

$\frac{lim_{x \to 6} x^{2} - lim_{x \to 6} 3 x - lim_{x \to 6} 18}{lim_{x \to 6} x^{2} - 12 x + 36}$

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

$\frac{{(lim_{x \to 6} x)}^{2} - lim_{x \to 6} 3 x - lim_{x \to 6} 18}{lim_{x \to 6} x^{2} - 12 x + 36}$

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

$\frac{{(lim_{x \to 6} x)}^{2} - 3 lim_{x \to 6} x - lim_{x \to 6} 18}{lim_{x \to 6} x^{2} - 12 x + 36}$

Evaluate the limit of $18$ which is constant as $x$ approaches $6$ .

$\frac{{(lim_{x \to 6} x)}^{2} - 3 lim_{x \to 6} x - 1 \cdot 18}{lim_{x \to 6} x^{2} - 12 x + 36}$

Evaluate the limits by plugging in $6$ for all occurrences of $x$ .

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$\frac{6^{2} - 3 lim_{x \to 6} x - 1 \cdot 18}{lim_{x \to 6} x^{2} - 12 x + 36}$

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$\frac{6^{2} - 3 \cdot 6 - 1 \cdot 18}{lim_{x \to 6} x^{2} - 12 x + 36}$

Simplify the answer.

Simplify each term.

Raise $6$ to the power of $2$ .

$\frac{36 - 3 \cdot 6 - 1 \cdot 18}{lim_{x \to 6} x^{2} - 12 x + 36}$

Multiply $- 3$ by $6$ .

$\frac{36 - 18 - 1 \cdot 18}{lim_{x \to 6} x^{2} - 12 x + 36}$

Multiply $- 1$ by $18$ .

$\frac{36 - 18 - 18}{lim_{x \to 6} x^{2} - 12 x + 36}$

Subtract $18$ from $36$ .

$\frac{18 - 18}{lim_{x \to 6} x^{2} - 12 x + 36}$

Subtract $18$ from $18$ .

$\frac{0}{lim_{x \to 6} x^{2} - 12 x + 36}$

Evaluate the limit of the denominator.

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $6$ .

$\frac{0}{lim_{x \to 6} x^{2} - lim_{x \to 6} 12 x + lim_{x \to 6} 36}$

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

$\frac{0}{{(lim_{x \to 6} x)}^{2} - lim_{x \to 6} 12 x + lim_{x \to 6} 36}$

Move the term $12$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{{(lim_{x \to 6} x)}^{2} - 12 lim_{x \to 6} x + lim_{x \to 6} 36}$

Evaluate the limit of $36$ which is constant as $x$ approaches $6$ .

$\frac{0}{{(lim_{x \to 6} x)}^{2} - 12 lim_{x \to 6} x + 36}$

Evaluate the limits by plugging in $6$ for all occurrences of $x$ .

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$\frac{0}{6^{2} - 12 lim_{x \to 6} x + 36}$

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$\frac{0}{6^{2} - 12 \cdot 6 + 36}$

Simplify the answer.

Simplify each term.

Raise $6$ to the power of $2$ .

$\frac{0}{36 - 12 \cdot 6 + 36}$

Multiply $- 12$ by $6$ .

$\frac{0}{36 - 72 + 36}$

Subtract $72$ from $36$ .

$\frac{0}{- 36 + 36}$

Add $- 36$ and $36$ .

$\frac{0}{0}$

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.

$lim_{x \to 6} \frac{x^{2} - 3 x - 18}{x^{2} - 12 x + 36} = lim_{x \to 6} \frac{\frac{d}{d x} [x^{2} - 3 x - 18]}{\frac{d}{d x} [x^{2} - 12 x + 36]}$

Find the derivative of the numerator and denominator.

Differentiate the numerator and denominator.

$lim_{x \to 6} \frac{\frac{d}{d x} [x^{2} - 3 x - 18]}{\frac{d}{d x} [x^{2} - 12 x + 36]}$

By the Sum Rule, the derivative of $x^{2} - 3 x - 18$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 18]$ .

$lim_{x \to 6} \frac{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 18]}{\frac{d}{d x} [x^{2} - 12 x + 36]}$

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

$lim_{x \to 6} \frac{2 x + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 18]}{\frac{d}{d x} [x^{2} - 12 x + 36]}$

Evaluate $\frac{d}{d x} [- 3 x]$ .

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$lim_{x \to 6} \frac{2 x - 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 18]}{\frac{d}{d x} [x^{2} - 12 x + 36]}$

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

$lim_{x \to 6} \frac{2 x - 3 \cdot 1 + \frac{d}{d x} [- 18]}{\frac{d}{d x} [x^{2} - 12 x + 36]}$

Multiply $- 3$ by $1$ .

$lim_{x \to 6} \frac{2 x - 3 + \frac{d}{d x} [- 18]}{\frac{d}{d x} [x^{2} - 12 x + 36]}$

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18$ with respect to $x$ is $0$ .

$lim_{x \to 6} \frac{2 x - 3 + 0}{\frac{d}{d x} [x^{2} - 12 x + 36]}$

Add $2 x - 3$ and $0$ .

$lim_{x \to 6} \frac{2 x - 3}{\frac{d}{d x} [x^{2} - 12 x + 36]}$

By the Sum Rule, the derivative of $x^{2} - 12 x + 36$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [36]$ .

$lim_{x \to 6} \frac{2 x - 3}{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [36]}$

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

$lim_{x \to 6} \frac{2 x - 3}{2 x + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [36]}$

Evaluate $\frac{d}{d x} [- 12 x]$ .

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x$ with respect to $x$ is $- 12 \frac{d}{d x} [x]$ .

$lim_{x \to 6} \frac{2 x - 3}{2 x - 12 \frac{d}{d x} [x] + \frac{d}{d x} [36]}$

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

$lim_{x \to 6} \frac{2 x - 3}{2 x - 12 \cdot 1 + \frac{d}{d x} [36]}$

Multiply $- 12$ by $1$ .

$lim_{x \to 6} \frac{2 x - 3}{2 x - 12 + \frac{d}{d x} [36]}$

Since $36$ is constant with respect to $x$ , the derivative of $36$ with respect to $x$ is $0$ .

$lim_{x \to 6} \frac{2 x - 3}{2 x - 12 + 0}$

Add $2 x - 12$ and $0$ .

$lim_{x \to 6} \frac{2 x - 3}{2 x - 12}$

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 6 of (x^2-3x-18)/(x^2-12x+36)

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