Calculus Examples

$\frac{lim_{x \to 2} 3 x^{2} - 5}{x^{2} + 3 x - 3}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{lim_{x \to 2} 3 x^{2} - lim_{x \to 2} 5}{x^{2} + 3 x - 3}$

Step 1.2

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

$\frac{3 lim_{x \to 2} x^{2} - lim_{x \to 2} 5}{x^{2} + 3 x - 3}$

Step 1.3

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

$\frac{3 {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} 5}{x^{2} + 3 x - 3}$

Step 1.4

Evaluate the limit of $5$ which is constant as $x$ approaches $2$ .

$\frac{3 {(lim_{x \to 2} x)}^{2} - 1 \cdot 5}{x^{2} + 3 x - 3}$

Step 2

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

$\frac{3 \cdot 2^{2} - 1 \cdot 5}{x^{2} + 3 x - 3}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{3 \cdot 4 - 1 \cdot 5}{x^{2} + 3 x - 3}$

Step 3.2

Multiply $3$ by $4$ .

$\frac{12 - 1 \cdot 5}{x^{2} + 3 x - 3}$

Step 3.3

Multiply $- 1$ by $5$ .

$\frac{12 - 5}{x^{2} + 3 x - 3}$

Step 3.4

Subtract $5$ from $12$ .

$\frac{7}{x^{2} + 3 x - 3}$