Simplify limit as x approaches 2 of (x^3-7x+6)/(3x^2-8x+4)

$lim_{x \to 2} \frac{x^{3} - 7 x + 6}{3 x^{2} - 8 x + 4}$

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.

$\frac{lim_{x \to 2} x^{3} - 7 x + 6}{lim_{x \to 2} 3 x^{2} - 8 x + 4}$

Evaluate the limit of the numerator.

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2} x^{3} - lim_{x \to 2} 7 x + lim_{x \to 2} 6}{lim_{x \to 2} 3 x^{2} - 8 x + 4}$

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 2} x)}^{3} - lim_{x \to 2} 7 x + lim_{x \to 2} 6}{lim_{x \to 2} 3 x^{2} - 8 x + 4}$

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

$\frac{{(lim_{x \to 2} x)}^{3} - 7 lim_{x \to 2} x + lim_{x \to 2} 6}{lim_{x \to 2} 3 x^{2} - 8 x + 4}$

Evaluate the limit of $6$ which is constant as $x$ approaches $2$ .

$\frac{{(lim_{x \to 2} x)}^{3} - 7 lim_{x \to 2} x + 6}{lim_{x \to 2} 3 x^{2} - 8 x + 4}$

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{2^{3} - 7 lim_{x \to 2} x + 6}{lim_{x \to 2} 3 x^{2} - 8 x + 4}$

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{2^{3} - 7 \cdot 2 + 6}{lim_{x \to 2} 3 x^{2} - 8 x + 4}$

Simplify the answer.

Simplify each term.

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

$\frac{8 - 7 \cdot 2 + 6}{lim_{x \to 2} 3 x^{2} - 8 x + 4}$

Multiply $- 7$ by $2$ .

$\frac{8 - 14 + 6}{lim_{x \to 2} 3 x^{2} - 8 x + 4}$

Subtract $14$ from $8$ .

$\frac{- 6 + 6}{lim_{x \to 2} 3 x^{2} - 8 x + 4}$

Add $- 6$ and $6$ .

$\frac{0}{lim_{x \to 2} 3 x^{2} - 8 x + 4}$

Evaluate the limit of the denominator.

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{0}{lim_{x \to 2} 3 x^{2} - lim_{x \to 2} 8 x + lim_{x \to 2} 4}$

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

$\frac{0}{3 lim_{x \to 2} x^{2} - lim_{x \to 2} 8 x + lim_{x \to 2} 4}$

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

$\frac{0}{3 {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} 8 x + lim_{x \to 2} 4}$

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{3 {(lim_{x \to 2} x)}^{2} - 8 lim_{x \to 2} x + lim_{x \to 2} 4}$

Evaluate the limit of $4$ which is constant as $x$ approaches $2$ .

$\frac{0}{3 {(lim_{x \to 2} x)}^{2} - 8 lim_{x \to 2} x + 4}$

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{0}{3 \cdot 2^{2} - 8 lim_{x \to 2} x + 4}$

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{0}{3 \cdot 2^{2} - 8 \cdot 2 + 4}$

Simplify the answer.

Simplify each term.

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

$\frac{0}{3 \cdot 4 - 8 \cdot 2 + 4}$

Multiply $3$ by $4$ .

$\frac{0}{12 - 8 \cdot 2 + 4}$

Multiply $- 8$ by $2$ .

$\frac{0}{12 - 16 + 4}$

Subtract $16$ from $12$ .

$\frac{0}{- 4 + 4}$

Add $- 4$ and $4$ .

$\frac{0}{0}$

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.

$lim_{x \to 2} \frac{x^{3} - 7 x + 6}{3 x^{2} - 8 x + 4} = lim_{x \to 2} \frac{\frac{d}{d x} [x^{3} - 7 x + 6]}{\frac{d}{d x} [3 x^{2} - 8 x + 4]}$

Find the derivative of the numerator and denominator.

Differentiate the numerator and denominator.

$lim_{x \to 2} \frac{\frac{d}{d x} [x^{3} - 7 x + 6]}{\frac{d}{d x} [3 x^{2} - 8 x + 4]}$

By the Sum Rule, the derivative of $x^{3} - 7 x + 6$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [6]$ .

$lim_{x \to 2} \frac{\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [6]}{\frac{d}{d x} [3 x^{2} - 8 x + 4]}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$lim_{x \to 2} \frac{3 x^{2} + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [6]}{\frac{d}{d x} [3 x^{2} - 8 x + 4]}$

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

$lim_{x \to 2} \frac{3 x^{2} - 7 \frac{d}{d x} [x] + \frac{d}{d x} [6]}{\frac{d}{d x} [3 x^{2} - 8 x + 4]}$

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

$lim_{x \to 2} \frac{3 x^{2} - 7 \cdot 1 + \frac{d}{d x} [6]}{\frac{d}{d x} [3 x^{2} - 8 x + 4]}$

Multiply $- 7$ by $1$ .

$lim_{x \to 2} \frac{3 x^{2} - 7 + \frac{d}{d x} [6]}{\frac{d}{d x} [3 x^{2} - 8 x + 4]}$

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$lim_{x \to 2} \frac{3 x^{2} - 7 + 0}{\frac{d}{d x} [3 x^{2} - 8 x + 4]}$

Add $3 x^{2} - 7$ and $0$ .

$lim_{x \to 2} \frac{3 x^{2} - 7}{\frac{d}{d x} [3 x^{2} - 8 x + 4]}$

By the Sum Rule, the derivative of $3 x^{2} - 8 x + 4$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [4]$ .

$lim_{x \to 2} \frac{3 x^{2} - 7}{\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [4]}$

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to 2} \frac{3 x^{2} - 7}{3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [4]}$

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

$lim_{x \to 2} \frac{3 x^{2} - 7}{3 (2 x) + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [4]}$

Multiply $2$ by $3$ .

$lim_{x \to 2} \frac{3 x^{2} - 7}{6 x + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [4]}$

Evaluate $\frac{d}{d x} [- 8 x]$ .

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$lim_{x \to 2} \frac{3 x^{2} - 7}{6 x - 8 \frac{d}{d x} [x] + \frac{d}{d x} [4]}$

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

$lim_{x \to 2} \frac{3 x^{2} - 7}{6 x - 8 \cdot 1 + \frac{d}{d x} [4]}$

Multiply $- 8$ by $1$ .

$lim_{x \to 2} \frac{3 x^{2} - 7}{6 x - 8 + \frac{d}{d x} [4]}$

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$lim_{x \to 2} \frac{3 x^{2} - 7}{6 x - 8 + 0}$

Add $6 x - 8$ and $0$ .

$lim_{x \to 2} \frac{3 x^{2} - 7}{6 x - 8}$

Step 2

Evaluate the limit.

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2} 3 x^{2} - 7}{lim_{x \to 2} 6 x - 8}$

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2} 3 x^{2} - lim_{x \to 2} 7}{lim_{x \to 2} 6 x - 8}$

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

$\frac{3 lim_{x \to 2} x^{2} - lim_{x \to 2} 7}{lim_{x \to 2} 6 x - 8}$

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

$\frac{3 {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} 7}{lim_{x \to 2} 6 x - 8}$

Evaluate the limit of $7$ which is constant as $x$ approaches $2$ .

$\frac{3 {(lim_{x \to 2} x)}^{2} - 1 \cdot 7}{lim_{x \to 2} 6 x - 8}$

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{3 {(lim_{x \to 2} x)}^{2} - 1 \cdot 7}{lim_{x \to 2} 6 x - lim_{x \to 2} 8}$

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 {(lim_{x \to 2} x)}^{2} - 1 \cdot 7}{6 lim_{x \to 2} x - lim_{x \to 2} 8}$

Evaluate the limit of $8$ which is constant as $x$ approaches $2$ .

$\frac{3 {(lim_{x \to 2} x)}^{2} - 1 \cdot 7}{6 lim_{x \to 2} x - 1 \cdot 8}$

Step 3

Simplify the numerator.

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{3 \cdot 2^{2} - 1 \cdot 7}{6 lim_{x \to 2} x - 1 \cdot 8}$

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

$\frac{3 \cdot 4 - 1 \cdot 7}{6 lim_{x \to 2} x - 1 \cdot 8}$

Multiply $3$ by $4$ .

$\frac{12 - 1 \cdot 7}{6 lim_{x \to 2} x - 1 \cdot 8}$

Multiply $- 1$ by $7$ .

$\frac{12 - 7}{6 lim_{x \to 2} x - 1 \cdot 8}$

Subtract $7$ from $12$ .

$\frac{5}{6 lim_{x \to 2} x - 1 \cdot 8}$

Step 4

Simplify the denominator.

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{5}{6 \cdot 2 - 1 \cdot 8}$

Multiply $6$ by $2$ .

$\frac{5}{12 - 1 \cdot 8}$

Multiply $- 1$ by $8$ .

$\frac{5}{12 - 8}$

Subtract $8$ from $12$ .

$\frac{5}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{4}$

Decimal Form:

$1.25$

Simplify limit as x approaches 2 of (x^3-7x+6)/(3x^2-8x+4)

Related Questions

Support LCMGCF.com by disabling your adblocker.