Calculus Examples

$\frac{lim_{x \to 0.5} 2 x^{2} + 3 x - 1}{4 x + 5}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0.5$ .

$\frac{lim_{x \to 0.5} 2 x^{2} + lim_{x \to 0.5} 3 x - lim_{x \to 0.5} 1}{4 x + 5}$

Step 2

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

$\frac{2 lim_{x \to 0.5} x^{2} + lim_{x \to 0.5} 3 x - lim_{x \to 0.5} 1}{4 x + 5}$

Step 3

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

$\frac{2 {(lim_{x \to 0.5} x)}^{2} + lim_{x \to 0.5} 3 x - lim_{x \to 0.5} 1}{4 x + 5}$

Step 4

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

$\frac{2 {(lim_{x \to 0.5} x)}^{2} + 3 lim_{x \to 0.5} x - lim_{x \to 0.5} 1}{4 x + 5}$

Step 5

Evaluate the limit of $1$ which is constant as $x$ approaches $0.5$ .

$\frac{2 {(lim_{x \to 0.5} x)}^{2} + 3 lim_{x \to 0.5} x - 1 \cdot 1}{4 x + 5}$

Step 6

Evaluate the limits by plugging in $0.5$ for all occurrences of $x$ .

Tap for more steps...

Step 6.1

Evaluate the limit of $x$ by plugging in $0.5$ for $x$ .

$\frac{2 \cdot {0.5}^{2} + 3 lim_{x \to 0.5} x - 1 \cdot 1}{4 x + 5}$

Step 6.2

Evaluate the limit of $x$ by plugging in $0.5$ for $x$ .

$\frac{2 \cdot {0.5}^{2} + 3 \cdot 0.5 - 1 \cdot 1}{4 x + 5}$

Step 7

Simplify the numerator.

Tap for more steps...

Step 7.1

Raise $0.5$ to the power of $2$ .

$\frac{2 \cdot 0.25 + 3 \cdot 0.5 - 1 \cdot 1}{4 x + 5}$

Step 7.2

Multiply $2$ by $0.25$ .

$\frac{0.5 + 3 \cdot 0.5 - 1 \cdot 1}{4 x + 5}$

Step 7.3

Multiply $3$ by $0.5$ .

$\frac{0.5 + 1.5 - 1 \cdot 1}{4 x + 5}$

Step 7.4

Multiply $- 1$ by $1$ .

$\frac{0.5 + 1.5 - 1}{4 x + 5}$

Step 7.5

Add $0.5$ and $1.5$ .

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

Step 7.6

Subtract $1$ from $2$ .

$\frac{1}{4 x + 5}$