Calculus Examples

$lim_{x \to - 1} x^{4} - 3 x^{3} + 2 x^{2} - 5 x - 7$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 1$ .

$lim_{x \to - 1} x^{4} - lim_{x \to - 1} 3 x^{3} + lim_{x \to - 1} 2 x^{2} - lim_{x \to - 1} 5 x - lim_{x \to - 1} 7$

Step 2

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

${(lim_{x \to - 1} x)}^{4} - lim_{x \to - 1} 3 x^{3} + lim_{x \to - 1} 2 x^{2} - lim_{x \to - 1} 5 x - lim_{x \to - 1} 7$

Step 3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

${(lim_{x \to - 1} x)}^{4} - 3 lim_{x \to - 1} x^{3} + lim_{x \to - 1} 2 x^{2} - lim_{x \to - 1} 5 x - lim_{x \to - 1} 7$

Step 4

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

${(lim_{x \to - 1} x)}^{4} - 3 {(lim_{x \to - 1} x)}^{3} + lim_{x \to - 1} 2 x^{2} - lim_{x \to - 1} 5 x - lim_{x \to - 1} 7$

Step 5

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

${(lim_{x \to - 1} x)}^{4} - 3 {(lim_{x \to - 1} x)}^{3} + 2 lim_{x \to - 1} x^{2} - lim_{x \to - 1} 5 x - lim_{x \to - 1} 7$

Step 6

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

${(lim_{x \to - 1} x)}^{4} - 3 {(lim_{x \to - 1} x)}^{3} + 2 {(lim_{x \to - 1} x)}^{2} - lim_{x \to - 1} 5 x - lim_{x \to - 1} 7$

Step 7

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

${(lim_{x \to - 1} x)}^{4} - 3 {(lim_{x \to - 1} x)}^{3} + 2 {(lim_{x \to - 1} x)}^{2} - 5 lim_{x \to - 1} x - lim_{x \to - 1} 7$

Step 8

Evaluate the limit of $7$ which is constant as $x$ approaches $- 1$ .

${(lim_{x \to - 1} x)}^{4} - 3 {(lim_{x \to - 1} x)}^{3} + 2 {(lim_{x \to - 1} x)}^{2} - 5 lim_{x \to - 1} x - 1 \cdot 7$

Step 9

Evaluate the limits by plugging in $- 1$ for all occurrences of $x$ .

Tap for more steps...

Step 9.1

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

${(- 1)}^{4} - 3 {(lim_{x \to - 1} x)}^{3} + 2 {(lim_{x \to - 1} x)}^{2} - 5 lim_{x \to - 1} x - 1 \cdot 7$

Step 9.2

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

${(- 1)}^{4} - 3 {(- 1)}^{3} + 2 {(lim_{x \to - 1} x)}^{2} - 5 lim_{x \to - 1} x - 1 \cdot 7$

Step 9.3

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

${(- 1)}^{4} - 3 {(- 1)}^{3} + 2 {(- 1)}^{2} - 5 lim_{x \to - 1} x - 1 \cdot 7$

Step 9.4

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

${(- 1)}^{4} - 3 {(- 1)}^{3} + 2 {(- 1)}^{2} - 5 \cdot - 1 - 1 \cdot 7$

Step 10

Simplify the answer.

Tap for more steps...

Step 10.1

Simplify each term.

Tap for more steps...

Step 10.1.1

Raise $- 1$ to the power of $4$ .

$1 - 3 {(- 1)}^{3} + 2 {(- 1)}^{2} - 5 \cdot - 1 - 1 \cdot 7$

Step 10.1.2

Raise $- 1$ to the power of $3$ .

$1 - 3 \cdot - 1 + 2 {(- 1)}^{2} - 5 \cdot - 1 - 1 \cdot 7$

Step 10.1.3

Multiply $- 3$ by $- 1$ .

$1 + 3 + 2 {(- 1)}^{2} - 5 \cdot - 1 - 1 \cdot 7$

Step 10.1.4

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

$1 + 3 + 2 \cdot 1 - 5 \cdot - 1 - 1 \cdot 7$

Step 10.1.5

Multiply $2$ by $1$ .

$1 + 3 + 2 - 5 \cdot - 1 - 1 \cdot 7$

Step 10.1.6

Multiply $- 5$ by $- 1$ .

$1 + 3 + 2 + 5 - 1 \cdot 7$

Step 10.1.7

Multiply $- 1$ by $7$ .

$1 + 3 + 2 + 5 - 7$

Step 10.2

Add $1$ and $3$ .

$4 + 2 + 5 - 7$

Step 10.3

Add $4$ and $2$ .

$6 + 5 - 7$

Step 10.4

Add $6$ and $5$ .

$11 - 7$

Step 10.5

Subtract $7$ from $11$ .

$4$