Calculus Examples

$lim_{h \to 2} \frac{(- 0.1 {(2 + h)}^{2} - (1 + h) + 32) - (- 0.1 {(2)}^{2} - 1 + 32)}{h}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $h$ approaches $2$ .

$\frac{lim_{h \to 2} (- 0.1 {(2 + h)}^{2} - (1 + h) + 32) - (- 0.1 {(2)}^{2} - 1 + 32)}{lim_{h \to 2} h}$

Step 2

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

Tap for more steps...

Step 2.1

Evaluate the limit of $(- 0.1 {(2 + h)}^{2} - (1 + h) + 32) - (- 0.1 {(2)}^{2} - 1 + 32)$ by plugging in $2$ for $h$ .

$\frac{- 0.1 {(2 + 2)}^{2} - (1 + 2) + 32 - (- 0.1 {(2)}^{2} - 1 + 32)}{lim_{h \to 2} h}$

Step 2.2

Simplify each term.

Tap for more steps...

Step 2.2.1

Add $2$ and $2$ .

$\frac{- 0.1 \cdot 4^{2} - (1 + 2) + 32 - (- 0.1 {(2)}^{2} - 1 + 32)}{lim_{h \to 2} h}$

Step 2.2.2

Raise $4$ to the power of $2$ .

$\frac{- 0.1 \cdot 16 - (1 + 2) + 32 - (- 0.1 {(2)}^{2} - 1 + 32)}{lim_{h \to 2} h}$

Step 2.2.3

Multiply $- 0.1$ by $16$ .

$\frac{- 1.6 - (1 + 2) + 32 - (- 0.1 {(2)}^{2} - 1 + 32)}{lim_{h \to 2} h}$

Step 2.2.4

Add $1$ and $2$ .

$\frac{- 1.6 - 1 \cdot 3 + 32 - (- 0.1 {(2)}^{2} - 1 + 32)}{lim_{h \to 2} h}$

Step 2.2.5

Multiply $- 1$ by $3$ .

$\frac{- 1.6 - 3 + 32 - (- 0.1 {(2)}^{2} - 1 + 32)}{lim_{h \to 2} h}$

Step 2.2.6

Simplify each term.

Tap for more steps...

Step 2.2.6.1

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

$\frac{- 1.6 - 3 + 32 - (- 0.1 \cdot 4 - 1 + 32)}{lim_{h \to 2} h}$

Step 2.2.6.2

Multiply $- 0.1$ by $4$ .

$\frac{- 1.6 - 3 + 32 - (- 0.4 - 1 + 32)}{lim_{h \to 2} h}$

Step 2.2.7

Subtract $1$ from $- 0.4$ .

$\frac{- 1.6 - 3 + 32 - (- 1.4 + 32)}{lim_{h \to 2} h}$

Step 2.2.8

Add $- 1.4$ and $32$ .

$\frac{- 1.6 - 3 + 32 - 1 \cdot 30.6}{lim_{h \to 2} h}$

Step 2.2.9

Multiply $- 1$ by $30.6$ .

$\frac{- 1.6 - 3 + 32 - 30.6}{lim_{h \to 2} h}$

Step 2.3

Subtract $3$ from $- 1.6$ .

$\frac{- 4.6 + 32 - 30.6}{lim_{h \to 2} h}$

Step 2.4

Add $- 4.6$ and $32$ .

$\frac{27.4 - 30.6}{lim_{h \to 2} h}$

Step 2.5

Subtract $30.6$ from $27.4$ .

$\frac{- 3.2}{lim_{h \to 2} h}$

Step 2.6

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

$\frac{- 3.2}{2}$

Step 3

Divide $- 3.2$ by $2$ .

$- 1.6$