Calculus Examples

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

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

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

Step 2

Move the term $\frac{1}{4}$ outside of the limit because it is constant with respect to $x$ .

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

Step 3

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

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

Step 4

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

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

Step 5

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

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

Step 5.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{1}{4} \cdot 4^{5} - 2 \cdot 4$

Step 6

Simplify the answer.

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Step 6.1

Simplify each term.

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Step 6.1.1

Cancel the common factor of $4$ .

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Step 6.1.1.1

Factor $4$ out of $4^{5}$ .

$\frac{1}{4} \cdot (4 \cdot 4^{4}) - 2 \cdot 4$

Step 6.1.1.2

Cancel the common factor.

$\frac{1}{4} \cdot (4 \cdot 4^{4}) - 2 \cdot 4$

Step 6.1.1.3

Rewrite the expression.

$4^{4} - 2 \cdot 4$

Step 6.1.2

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

$256 - 2 \cdot 4$

Step 6.1.3

Multiply $- 2$ by $4$ .

$256 - 8$

Step 6.2

Subtract $8$ from $256$ .

$248$