Calculus Examples

$\frac{lim_{x \to 0} x^{2} - 6 x + 9}{x^{2}}$

Step 1

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

$\frac{lim_{x \to 0} x^{2} - lim_{x \to 0} 6 x + lim_{x \to 0} 9}{x^{2}}$

Step 2

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

$\frac{{(lim_{x \to 0} x)}^{2} - lim_{x \to 0} 6 x + lim_{x \to 0} 9}{x^{2}}$

Step 3

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

$\frac{{(lim_{x \to 0} x)}^{2} - 6 lim_{x \to 0} x + lim_{x \to 0} 9}{x^{2}}$

Step 4

Evaluate the limit of $9$ which is constant as $x$ approaches $0$ .

$\frac{{(lim_{x \to 0} x)}^{2} - 6 lim_{x \to 0} x + 9}{x^{2}}$

Step 5

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 5.1

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

$\frac{0^{2} - 6 lim_{x \to 0} x + 9}{x^{2}}$

Step 5.2

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

$\frac{0^{2} - 6 \cdot 0 + 9}{x^{2}}$

Step 6

Simplify the numerator.

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Step 6.1

Raising $0$ to any positive power yields $0$ .

$\frac{0 - 6 \cdot 0 + 9}{x^{2}}$

Step 6.2

Multiply $- 6$ by $0$ .

$\frac{0 + 0 + 9}{x^{2}}$

Step 6.3

Add $0$ and $0$ .

$\frac{0 + 9}{x^{2}}$

Step 6.4

Add $0$ and $9$ .

$\frac{9}{x^{2}}$