Calculus Examples

$lim_{x \to 0} \frac{2 x^{2} - 3 x + 4}{x} + \frac{5 x - 4}{x}$

Step 1

Combine fractions using a common denominator.

Tap for more steps...

Step 1.1

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{2 x^{2} - 3 x + 4 + 5 x - 4}{x}$

Step 1.2

Simplify the numerator.

Tap for more steps...

Step 1.2.1

Add $- 3 x$ and $5 x$ .

$lim_{x \to 0} \frac{2 x^{2} + 2 x + 4 - 4}{x}$

Step 1.2.2

Subtract $4$ from $4$ .

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

Step 1.2.3

Add $2 x^{2} + 2 x$ and $0$ .

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

Step 1.2.4

Factor $2 x$ out of $2 x^{2} + 2 x$ .

Tap for more steps...

Step 1.2.4.1

Factor $2 x$ out of $2 x^{2}$ .

$lim_{x \to 0} \frac{2 x (x) + 2 x}{x}$

Step 1.2.4.2

Factor $2 x$ out of $2 x$ .

$lim_{x \to 0} \frac{2 x (x) + 2 x (1)}{x}$

Step 1.2.4.3

Factor $2 x$ out of $2 x (x) + 2 x (1)$ .

$lim_{x \to 0} \frac{2 x (x + 1)}{x}$

Step 1.3

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.3.1

Cancel the common factor.

$lim_{x \to 0} \frac{2 x (x + 1)}{x}$

Step 1.3.2

Divide $2 (x + 1)$ by $1$ .

$lim_{x \to 0} 2 (x + 1)$

Step 1.4

Apply the distributive property.

$lim_{x \to 0} 2 x + 2 \cdot 1$

Step 1.5

Multiply $2$ by $1$ .

$lim_{x \to 0} 2 x + 2$

Step 2

Evaluate the limit.

Tap for more steps...

Step 2.1

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

$lim_{x \to 0} 2 x + lim_{x \to 0} 2$

Step 2.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$2 lim_{x \to 0} x + lim_{x \to 0} 2$

Step 2.3

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

$2 lim_{x \to 0} x + 2$

Step 3

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

$2 \cdot 0 + 2$

Step 4

Simplify the answer.

Tap for more steps...

Step 4.1

Multiply $2$ by $0$ .

$0 + 2$

Step 4.2

Add $0$ and $2$ .

$2$