Calculus Examples

$\frac{lim_{x \to 1} 3 x^{2} + 7 x - 2}{3 x^{2} - 4 x - 2}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{lim_{x \to 1} 3 x^{2} + lim_{x \to 1} 7 x - lim_{x \to 1} 2}{3 x^{2} - 4 x - 2}$

Step 2

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

$\frac{3 lim_{x \to 1} x^{2} + lim_{x \to 1} 7 x - lim_{x \to 1} 2}{3 x^{2} - 4 x - 2}$

Step 3

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

$\frac{3 {(lim_{x \to 1} x)}^{2} + lim_{x \to 1} 7 x - lim_{x \to 1} 2}{3 x^{2} - 4 x - 2}$

Step 4

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 {(lim_{x \to 1} x)}^{2} + 7 lim_{x \to 1} x - lim_{x \to 1} 2}{3 x^{2} - 4 x - 2}$

Step 5

Evaluate the limit of $2$ which is constant as $x$ approaches $1$ .

$\frac{3 {(lim_{x \to 1} x)}^{2} + 7 lim_{x \to 1} x - 1 \cdot 2}{3 x^{2} - 4 x - 2}$

Step 6

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

Tap for more steps...

Step 6.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{3 \cdot 1^{2} + 7 lim_{x \to 1} x - 1 \cdot 2}{3 x^{2} - 4 x - 2}$

Step 6.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{3 \cdot 1^{2} + 7 \cdot 1 - 1 \cdot 2}{3 x^{2} - 4 x - 2}$

Step 7

Simplify the numerator.

Tap for more steps...

Step 7.1

One to any power is one.

$\frac{3 \cdot 1 + 7 \cdot 1 - 1 \cdot 2}{3 x^{2} - 4 x - 2}$

Step 7.2

Multiply $3$ by $1$ .

$\frac{3 + 7 \cdot 1 - 1 \cdot 2}{3 x^{2} - 4 x - 2}$

Step 7.3

Multiply $7$ by $1$ .

$\frac{3 + 7 - 1 \cdot 2}{3 x^{2} - 4 x - 2}$

Step 7.4

Multiply $- 1$ by $2$ .

$\frac{3 + 7 - 2}{3 x^{2} - 4 x - 2}$

Step 7.5

Add $3$ and $7$ .

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

Step 7.6

Subtract $2$ from $10$ .

$\frac{8}{3 x^{2} - 4 x - 2}$