Calculus Examples

$\frac{lim_{x \to 2} x^{2} + 3 x - 10}{3 x^{2} - 5 x - 2}$

Step 1

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

$\frac{lim_{x \to 2} x^{2} + lim_{x \to 2} 3 x - lim_{x \to 2} 10}{3 x^{2} - 5 x - 2}$

Step 2

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

$\frac{{(lim_{x \to 2} x)}^{2} + lim_{x \to 2} 3 x - lim_{x \to 2} 10}{3 x^{2} - 5 x - 2}$

Step 3

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

$\frac{{(lim_{x \to 2} x)}^{2} + 3 lim_{x \to 2} x - lim_{x \to 2} 10}{3 x^{2} - 5 x - 2}$

Step 4

Evaluate the limit of $10$ which is constant as $x$ approaches $2$ .

$\frac{{(lim_{x \to 2} x)}^{2} + 3 lim_{x \to 2} x - 1 \cdot 10}{3 x^{2} - 5 x - 2}$

Step 5

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 5.1

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

$\frac{2^{2} + 3 lim_{x \to 2} x - 1 \cdot 10}{3 x^{2} - 5 x - 2}$

Step 5.2

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

$\frac{2^{2} + 3 \cdot 2 - 1 \cdot 10}{3 x^{2} - 5 x - 2}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$\frac{4 + 3 \cdot 2 - 1 \cdot 10}{3 x^{2} - 5 x - 2}$

Step 6.1.2

Multiply $3$ by $2$ .

$\frac{4 + 6 - 1 \cdot 10}{3 x^{2} - 5 x - 2}$

Step 6.1.3

Multiply $- 1$ by $10$ .

$\frac{4 + 6 - 10}{3 x^{2} - 5 x - 2}$

Step 6.1.4

Add $4$ and $6$ .

$\frac{10 - 10}{3 x^{2} - 5 x - 2}$

Step 6.1.5

Subtract $10$ from $10$ .

$\frac{0}{3 x^{2} - 5 x - 2}$

Step 6.2

Factor by grouping.

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Step 6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = - 5$ .

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Step 6.2.1.1

Factor $- 5$ out of $- 5 x$ .

$\frac{0}{3 x^{2} - 5 (x) - 2}$

Step 6.2.1.2

Rewrite $- 5$ as $1$ plus $- 6$

$\frac{0}{3 x^{2} + (1 - 6) x - 2}$

Step 6.2.1.3

Apply the distributive property.

$\frac{0}{3 x^{2} + 1 x - 6 x - 2}$

Step 6.2.1.4

Multiply $x$ by $1$ .

$\frac{0}{3 x^{2} + x - 6 x - 2}$

Step 6.2.2

Factor out the greatest common factor from each group.

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Step 6.2.2.1

Group the first two terms and the last two terms.

$\frac{0}{(3 x^{2} + x) - 6 x - 2}$

Step 6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{0}{x (3 x + 1) - 2 (3 x + 1)}$

Step 6.2.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

$\frac{0}{(3 x + 1) (x - 2)}$

Step 6.3

Divide $0$ by $(3 x + 1) (x - 2)$ .

$0$