Calculus Examples

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

Step 1

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

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

Step 2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $8$ .

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

Step 3

Move the exponent $2$ from ${(4 x - 3)}^{2}$ outside the limit using the Limits Power Rule.

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

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

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

Step 5

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 6

Evaluate the limit of $3$ which is constant as $x$ approaches $8$ .

$\frac{2 {(4 lim_{x \to 8} x - 1 \cdot 3)}^{2} \cdot lim_{x \to 8} x - 4}{7 x^{2 (- x + 7)}}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{2 {(4 lim_{x \to 8} x - 1 \cdot 3)}^{2} \cdot (lim_{x \to 8} x - lim_{x \to 8} 4)}{7 x^{2 (- x + 7)}}$

Step 8

Evaluate the limit of $4$ which is constant as $x$ approaches $8$ .

$\frac{2 {(4 lim_{x \to 8} x - 1 \cdot 3)}^{2} \cdot (lim_{x \to 8} x - 1 \cdot 4)}{7 x^{2 (- x + 7)}}$

Step 9

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{2 {(4 \cdot 8 - 1 \cdot 3)}^{2} \cdot (lim_{x \to 8} x - 1 \cdot 4)}{7 x^{2 (- x + 7)}}$

Step 9.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{2 {(4 \cdot 8 - 1 \cdot 3)}^{2} \cdot (8 - 1 \cdot 4)}{7 x^{2 (- x + 7)}}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Combine exponents.

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Step 10.1.1.1

Multiply $4$ by $8$ .

$\frac{2 {(32 - 1 \cdot 3)}^{2} (8 - 1 \cdot 4)}{7 x^{2 (- x + 7)}}$

Step 10.1.1.2

Multiply $- 1$ by $3$ .

$\frac{2 {(32 - 3)}^{2} (8 - 1 \cdot 4)}{7 x^{2 (- x + 7)}}$

Step 10.1.1.3

Multiply $- 1$ by $4$ .

$\frac{2 {(32 - 3)}^{2} (8 - 4)}{7 x^{2 (- x + 7)}}$

Step 10.1.2

Subtract $3$ from $32$ .

$\frac{2 \cdot 29^{2} (8 - 4)}{7 x^{2 (- x + 7)}}$

Step 10.1.3

Subtract $4$ from $8$ .

$\frac{2 \cdot 29^{2} \cdot 4}{7 x^{2 (- x + 7)}}$

Step 10.1.4

Multiply $4$ by $2$ .

$\frac{8 \cdot 29^{2}}{7 x^{2 (- x + 7)}}$

Step 10.1.5

Raise $29$ to the power of $2$ .

$\frac{8 \cdot 841}{7 x^{2 (- x + 7)}}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Apply the distributive property.

$\frac{8 \cdot 841}{7 x^{2 (- x) + 2 \cdot 7}}$

Step 10.2.2

Multiply $- 1$ by $2$ .

$\frac{8 \cdot 841}{7 x^{- 2 x + 2 \cdot 7}}$

Step 10.2.3

Multiply $2$ by $7$ .

$\frac{8 \cdot 841}{7 x^{- 2 x + 14}}$

Step 10.3

Multiply $8$ by $841$ .

$\frac{6728}{7 x^{- 2 x + 14}}$