Simplify limit as x approaches 3 of (x^2-7x+12)/(x-3)

$lim_{x \to 3} \frac{x^{2} - 7 x + 12}{x - 3}$

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.

$\frac{lim_{x \to 3} x^{2} - 7 x + 12}{lim_{x \to 3} x - 3}$

Evaluate the limit of the numerator.

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} x^{2} - lim_{x \to 3} 7 x + lim_{x \to 3} 12}{lim_{x \to 3} x - 3}$

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

$\frac{{(lim_{x \to 3} x)}^{2} - lim_{x \to 3} 7 x + lim_{x \to 3} 12}{lim_{x \to 3} x - 3}$

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 3} x)}^{2} - 7 lim_{x \to 3} x + lim_{x \to 3} 12}{lim_{x \to 3} x - 3}$

Evaluate the limit of $12$ which is constant as $x$ approaches $3$ .

$\frac{{(lim_{x \to 3} x)}^{2} - 7 lim_{x \to 3} x + 12}{lim_{x \to 3} x - 3}$

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{3^{2} - 7 lim_{x \to 3} x + 12}{lim_{x \to 3} x - 3}$

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{3^{2} - 7 \cdot 3 + 12}{lim_{x \to 3} x - 3}$

Simplify the answer.

Simplify each term.

Raise $3$ to the power of $2$ .

$\frac{9 - 7 \cdot 3 + 12}{lim_{x \to 3} x - 3}$

Multiply $- 7$ by $3$ .

$\frac{9 - 21 + 12}{lim_{x \to 3} x - 3}$

Subtract $21$ from $9$ .

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

Add $- 12$ and $12$ .

$\frac{0}{lim_{x \to 3} x - 3}$

Evaluate the limit of the denominator.

Evaluate the limit.

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{0}{lim_{x \to 3} x - lim_{x \to 3} 3}$

Evaluate the limit of $3$ which is constant as $x$ approaches $3$ .

$\frac{0}{lim_{x \to 3} x - 1 \cdot 3}$

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{0}{3 - 1 \cdot 3}$

Simplify the answer.

Multiply $- 1$ by $3$ .

$\frac{0}{3 - 3}$

Subtract $3$ from $3$ .

$\frac{0}{0}$

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.

$lim_{x \to 3} \frac{x^{2} - 7 x + 12}{x - 3} = lim_{x \to 3} \frac{\frac{d}{d x} [x^{2} - 7 x + 12]}{\frac{d}{d x} [x - 3]}$

Find the derivative of the numerator and denominator.

Differentiate the numerator and denominator.

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{2} - 7 x + 12]}{\frac{d}{d x} [x - 3]}$

By the Sum Rule, the derivative of $x^{2} - 7 x + 12$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [12]$ .

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [12]}{\frac{d}{d x} [x - 3]}$

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

$lim_{x \to 3} \frac{2 x + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [12]}{\frac{d}{d x} [x - 3]}$

Evaluate $\frac{d}{d x} [- 7 x]$ .

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$lim_{x \to 3} \frac{2 x - 7 \frac{d}{d x} [x] + \frac{d}{d x} [12]}{\frac{d}{d x} [x - 3]}$

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

$lim_{x \to 3} \frac{2 x - 7 \cdot 1 + \frac{d}{d x} [12]}{\frac{d}{d x} [x - 3]}$

Multiply $- 7$ by $1$ .

$lim_{x \to 3} \frac{2 x - 7 + \frac{d}{d x} [12]}{\frac{d}{d x} [x - 3]}$

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$lim_{x \to 3} \frac{2 x - 7 + 0}{\frac{d}{d x} [x - 3]}$

Add $2 x - 7$ and $0$ .

$lim_{x \to 3} \frac{2 x - 7}{\frac{d}{d x} [x - 3]}$

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

$lim_{x \to 3} \frac{2 x - 7}{\frac{d}{d x} [x] + \frac{d}{d x} [- 3]}$

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

$lim_{x \to 3} \frac{2 x - 7}{1 + \frac{d}{d x} [- 3]}$

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$lim_{x \to 3} \frac{2 x - 7}{1 + 0}$

Add $1$ and $0$ .

$lim_{x \to 3} \frac{2 x - 7}{1}$

Divide $2 x - 7$ by $1$ .

$lim_{x \to 3} 2 x - 7$

Evaluate the limit.

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$lim_{x \to 3} 2 x - lim_{x \to 3} 7$

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

$2 lim_{x \to 3} x - lim_{x \to 3} 7$

Evaluate the limit of $7$ which is constant as $x$ approaches $3$ .

$2 lim_{x \to 3} x - 1 \cdot 7$

Simplify terms.

Simplify each term.

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$2 \cdot 3 - 1 \cdot 7$

Multiply $2$ by $3$ .

$6 - 1 \cdot 7$

Multiply $- 1$ by $7$ .

$6 - 7$

Subtract $7$ from $6$ .

$- 1$

Simplify limit as x approaches 3 of (x^2-7x+12)/(x-3)

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