Calculus Examples

$\frac{lim_{y \to 0} - 6 {(x - y)}^{2} - {(- 6 x)}^{2}}{y}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $0$ .

$\frac{- lim_{y \to 0} 6 {(x - y)}^{2} - lim_{y \to 0} {(- 6 x)}^{2}}{y}$

Step 2

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

$\frac{- 6 lim_{y \to 0} {(x - y)}^{2} - lim_{y \to 0} {(- 6 x)}^{2}}{y}$

Step 3

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

$\frac{- 6 {(lim_{y \to 0} x - y)}^{2} - lim_{y \to 0} {(- 6 x)}^{2}}{y}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $0$ .

$\frac{- 6 {(lim_{y \to 0} x - lim_{y \to 0} y)}^{2} - lim_{y \to 0} {(- 6 x)}^{2}}{y}$

Step 5

Evaluate the limit of $x$ which is constant as $y$ approaches $0$ .

$\frac{- 6 {(x - lim_{y \to 0} y)}^{2} - lim_{y \to 0} {(- 6 x)}^{2}}{y}$

Step 6

Evaluate the limit of ${(- 6 x)}^{2}$ which is constant as $y$ approaches $0$ .

$\frac{- 6 {(x - lim_{y \to 0} y)}^{2} - {(- 6 x)}^{2}}{y}$

Step 7

Simplify terms.

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Step 7.1

Evaluate the limit of $y$ by plugging in $0$ for $y$ .

$\frac{- 6 {(x + 0)}^{2} - {(- 6 x)}^{2}}{y}$

Step 7.2

Simplify the answer.

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Step 7.2.1

Simplify the numerator.

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Step 7.2.1.1

Add $x$ and $0$ .

$\frac{- 6 x^{2} - {(- 6 x)}^{2}}{y}$

Step 7.2.1.2

Apply the product rule to $- 6 x$ .

$\frac{- 6 x^{2} - ({(- 6)}^{2} x^{2})}{y}$

Step 7.2.1.3

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

$\frac{- 6 x^{2} - (36 x^{2})}{y}$

Step 7.2.1.4

Multiply $36$ by $- 1$ .

$\frac{- 6 x^{2} - 36 x^{2}}{y}$

Step 7.2.1.5

Subtract $36 x^{2}$ from $- 6 x^{2}$ .

$\frac{- 42 x^{2}}{y}$

Step 7.2.2

Move the negative in front of the fraction.

$- \frac{42 x^{2}}{y}$