Calculus Examples

$lim_{x \to 5} 5 (f {(x)}^{3}) + 7 (f (x) g (x))$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$lim_{x \to 5} 5 (f {(x)}^{3}) + lim_{x \to 5} 7 (f (x) g (x))$

Step 2

Move the term $5 f$ outside of the limit because it is constant with respect to $x$ .

$5 f lim_{x \to 5} {(x)}^{3} + lim_{x \to 5} 7 (f (x) g (x))$

Step 3

Move the exponent $3$ from ${(x)}^{3}$ outside the limit using the Limits Power Rule.

$5 f {(lim_{x \to 5} x)}^{3} + lim_{x \to 5} 7 (f (x) g (x))$

Step 4

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

$5 f {(lim_{x \to 5} x)}^{3} + 7 lim_{x \to 5} f (x) g (x)$

Step 5

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $5$ .

$5 f {(lim_{x \to 5} x)}^{3} + 7 lim_{x \to 5} f (x) \cdot lim_{x \to 5} g (x)$

Step 6

Evaluate the limits by plugging in $5$ for all occurrences of $x$ .

Tap for more steps...

Step 6.1

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$5 f \cdot 5^{3} + 7 lim_{x \to 5} f (x) \cdot lim_{x \to 5} g (x)$

Step 6.2

Evaluate the limit of $f (x)$ by plugging in $5$ for $x$ .

$5 f \cdot 5^{3} + 7 f (5) \cdot lim_{x \to 5} g (x)$

Step 6.3

Evaluate the limit of $g (x)$ by plugging in $5$ for $x$ .

$5 f \cdot 5^{3} + 7 f (5) \cdot g (5)$

Step 7

Simplify each term.

Tap for more steps...

Step 7.1

Multiply $5$ by $5^{3}$ by adding the exponents.

Tap for more steps...

Step 7.1.1

Move $5^{3}$ .

$5^{3} \cdot 5 f + 7 f (5) \cdot g (5)$

Step 7.1.2

Multiply $5^{3}$ by $5$ .

Tap for more steps...

Step 7.1.2.1

Raise $5$ to the power of $1$ .

$5^{3} \cdot 5^{1} f + 7 f (5) \cdot g (5)$

Step 7.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5^{3 + 1} f + 7 f (5) \cdot g (5)$

Step 7.1.3

Add $3$ and $1$ .

$5^{4} f + 7 f (5) \cdot g (5)$

Step 7.2

Raise $5$ to the power of $4$ .

$625 f + 7 f (5) g (5)$