Calculus Examples

$lim_{x \to 0} \frac{3}{x^{2} - 15 x + 2}$

Step 1

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

$3 lim_{x \to 0} \frac{1}{x^{2} - 15 x + 2}$

Step 2

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

$3 \frac{lim_{x \to 0} 1}{lim_{x \to 0} x^{2} - 15 x + 2}$

Step 3

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$3 \frac{1}{lim_{x \to 0} x^{2} - 15 x + 2}$

Step 4

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

$3 \frac{1}{lim_{x \to 0} x^{2} - lim_{x \to 0} 15 x + lim_{x \to 0} 2}$

Step 5

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

$3 \frac{1}{{(lim_{x \to 0} x)}^{2} - lim_{x \to 0} 15 x + lim_{x \to 0} 2}$

Step 6

Move the term $15$ outside of the limit because it is constant with respect to $x$ .

$3 \frac{1}{{(lim_{x \to 0} x)}^{2} - 15 lim_{x \to 0} x + lim_{x \to 0} 2}$

Step 7

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$3 \frac{1}{{(lim_{x \to 0} x)}^{2} - 15 lim_{x \to 0} x + 2}$

Step 8

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

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Step 8.1

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

$3 \frac{1}{0^{2} - 15 lim_{x \to 0} x + 2}$

Step 8.2

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

$3 \frac{1}{0^{2} - 15 \cdot 0 + 2}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the denominator.

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Step 9.1.1

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

$3 \frac{1}{0 - 15 \cdot 0 + 2}$

Step 9.1.2

Multiply $- 15$ by $0$ .

$3 \frac{1}{0 + 0 + 2}$

Step 9.1.3

Add $0$ and $0$ .

$3 \frac{1}{0 + 2}$

Step 9.1.4

Add $0$ and $2$ .

$3 (\frac{1}{2})$

Step 9.2

Combine $3$ and $\frac{1}{2}$ .

$\frac{3}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{2}$

Decimal Form:

$1.5$