Calculus Examples

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

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

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

Step 6

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 7

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

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

Step 8

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

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

Step 9

Evaluate the limit of $5$ which is constant as $x$ approaches $2$ .

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

Step 10

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

Tap for more steps...

Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

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

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

Step 11

Simplify the answer.

Tap for more steps...

Step 11.1

Simplify each term.

Tap for more steps...

Step 11.1.1

Multiply $2$ by $2^{5}$ by adding the exponents.

Tap for more steps...

Step 11.1.1.1

Multiply $2$ by $2^{5}$ .

Tap for more steps...

Step 11.1.1.1.1

Raise $2$ to the power of $1$ .

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

Step 11.1.1.1.2

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

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

Step 11.1.1.2

Add $1$ and $5$ .

$2^{6} - 2 \cdot 2^{4} + 4 \cdot 2^{3} + 2^{2} - 1 \cdot 5$

Step 11.1.2

Raise $2$ to the power of $6$ .

$64 - 2 \cdot 2^{4} + 4 \cdot 2^{3} + 2^{2} - 1 \cdot 5$

Step 11.1.3

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

$64 - 2 \cdot 16 + 4 \cdot 2^{3} + 2^{2} - 1 \cdot 5$

Step 11.1.4

Multiply $- 2$ by $16$ .

$64 - 32 + 4 \cdot 2^{3} + 2^{2} - 1 \cdot 5$

Step 11.1.5

Raise $2$ to the power of $3$ .

$64 - 32 + 4 \cdot 8 + 2^{2} - 1 \cdot 5$

Step 11.1.6

Multiply $4$ by $8$ .

$64 - 32 + 32 + 2^{2} - 1 \cdot 5$

Step 11.1.7

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

$64 - 32 + 32 + 4 - 1 \cdot 5$

Step 11.1.8

Multiply $- 1$ by $5$ .

$64 - 32 + 32 + 4 - 5$

Step 11.2

Subtract $32$ from $64$ .

$32 + 32 + 4 - 5$

Step 11.3

Add $32$ and $32$ .

$64 + 4 - 5$

Step 11.4

Add $64$ and $4$ .

$68 - 5$

Step 11.5

Subtract $5$ from $68$ .

$63$