Simplify limit as x approaches 3 of (x^2-9)/(x-3)

$lim_{x \to 3} \frac{x^{2} - 9}{x - 3}$

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.

$\frac{lim_{x \to 3} x^{2} - 9}{lim_{x \to 3} x - 3}$

Evaluate the limit of the numerator.

Evaluate the limit.

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

$\frac{lim_{x \to 3} x^{2} - lim_{x \to 3} 9}{lim_{x \to 3} x - 3}$

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

$\frac{{(lim_{x \to 3} x)}^{2} - lim_{x \to 3} 9}{lim_{x \to 3} x - 3}$

Evaluate the limit of $9$ which is constant as $x$ approaches $3$ .

$\frac{{(lim_{x \to 3} x)}^{2} - 1 \cdot 9}{lim_{x \to 3} x - 3}$

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

$\frac{3^{2} - 1 \cdot 9}{lim_{x \to 3} x - 3}$

Simplify the answer.

Simplify each term.

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

$\frac{9 - 1 \cdot 9}{lim_{x \to 3} x - 3}$

Multiply $- 1$ by $9$ .

$\frac{9 - 9}{lim_{x \to 3} x - 3}$

Subtract $9$ from $9$ .

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

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.

$lim_{x \to 3} \frac{x^{2} - 9}{x - 3} = lim_{x \to 3} \frac{\frac{d}{d x} [x^{2} - 9]}{\frac{d}{d x} [x - 3]}$

Find the derivative of the numerator and denominator.

Differentiate the numerator and denominator.

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{2} - 9]}{\frac{d}{d x} [x - 3]}$

By the Sum Rule, the derivative of $x^{2} - 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9]$ .

$lim_{x \to 3} \frac{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9]}{\frac{d}{d x} [x - 3]}$

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

$lim_{x \to 3} \frac{2 x + \frac{d}{d x} [- 9]}{\frac{d}{d x} [x - 3]}$

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

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

Add $2 x$ and $0$ .

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

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

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

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

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

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

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

Add $1$ and $0$ .

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

Divide $2 x$ by $1$ .

$lim_{x \to 3} 2 x$

Step 2

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

$2 lim_{x \to 3} x$

Step 3

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

$2 \cdot 3$

Step 4

Multiply $2$ by $3$ .

$6$

Simplify limit as x approaches 3 of (x^2-9)/(x-3)

Related Questions

Support LCMGCF.com by disabling your adblocker.