Calculus Examples

$f (x) = - 5 x lim_{h \to 0} f (x t h) - f (x)$

Step 1

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

$f (x) = - 5 x (lim_{h \to 0} f (x t h) - lim_{h \to 0} f (x))$

Step 2

Evaluate the limit of $f (x)$ which is constant as $h$ approaches $0$ .

$f (x) = - 5 x (lim_{h \to 0} f (x t h) - f (x))$

Step 3

Evaluate the limits by plugging in $0$ for all occurrences of $h$ .

Tap for more steps...

Step 3.1

Evaluate the limit of $f (x t h)$ by plugging in $0$ for $h$ .

$f (x) = - 5 x (f (x t \cdot 0) - f (x))$

Step 3.2

Multiply $x t \cdot 0$ .

Tap for more steps...

Step 3.2.1

Multiply $0$ by $x$ .

$f (x) = - 5 x (f (0 t) - f (x))$

Step 3.2.2

Multiply $0$ by $t$ .

$f (x) = - 5 x (f (0) - f (x))$

Step 4

Simplify the answer.

Tap for more steps...

Step 4.1

Apply the distributive property.

$f (x) = - 5 x f (0) - 5 x (- f (x))$

Step 4.2

Multiply $- 1$ by $- 5$ .

$f (x) = - 5 x f (0) + 5 x f (x)$