Calculus Examples

$lim_{x \to 7} - 7 x^{3} - 3 x^{2} + 8 x + 21$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $7$ .

$- lim_{x \to 7} 7 x^{3} - lim_{x \to 7} 3 x^{2} + lim_{x \to 7} 8 x + lim_{x \to 7} 21$

Step 2

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

$- 7 lim_{x \to 7} x^{3} - lim_{x \to 7} 3 x^{2} + lim_{x \to 7} 8 x + lim_{x \to 7} 21$

Step 3

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

$- 7 {(lim_{x \to 7} x)}^{3} - lim_{x \to 7} 3 x^{2} + lim_{x \to 7} 8 x + lim_{x \to 7} 21$

Step 4

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

$- 7 {(lim_{x \to 7} x)}^{3} - 3 lim_{x \to 7} x^{2} + lim_{x \to 7} 8 x + lim_{x \to 7} 21$

Step 5

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

$- 7 {(lim_{x \to 7} x)}^{3} - 3 {(lim_{x \to 7} x)}^{2} + lim_{x \to 7} 8 x + lim_{x \to 7} 21$

Step 6

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$- 7 {(lim_{x \to 7} x)}^{3} - 3 {(lim_{x \to 7} x)}^{2} + 8 lim_{x \to 7} x + lim_{x \to 7} 21$

Step 7

Evaluate the limit of $21$ which is constant as $x$ approaches $7$ .

$- 7 {(lim_{x \to 7} x)}^{3} - 3 {(lim_{x \to 7} x)}^{2} + 8 lim_{x \to 7} x + 21$

Step 8

Evaluate the limits by plugging in $7$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

$- 7 \cdot 7^{3} - 3 {(lim_{x \to 7} x)}^{2} + 8 lim_{x \to 7} x + 21$

Step 8.2

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

$- 7 \cdot 7^{3} - 3 \cdot 7^{2} + 8 lim_{x \to 7} x + 21$

Step 8.3

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

$- 7 \cdot 7^{3} - 3 \cdot 7^{2} + 8 \cdot 7 + 21$

Step 9

Simplify the answer.

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Step 9.1

Simplify each term.

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Step 9.1.1

Raise $7$ to the power of $3$ .

$- 7 \cdot 343 - 3 \cdot 7^{2} + 8 \cdot 7 + 21$

Step 9.1.2

Multiply $- 7$ by $343$ .

$- 2401 - 3 \cdot 7^{2} + 8 \cdot 7 + 21$

Step 9.1.3

Raise $7$ to the power of $2$ .

$- 2401 - 3 \cdot 49 + 8 \cdot 7 + 21$

Step 9.1.4

Multiply $- 3$ by $49$ .

$- 2401 - 147 + 8 \cdot 7 + 21$

Step 9.1.5

Multiply $8$ by $7$ .

$- 2401 - 147 + 56 + 21$

Step 9.2

Subtract $147$ from $- 2401$ .

$- 2548 + 56 + 21$

Step 9.3

Add $- 2548$ and $56$ .

$- 2492 + 21$

Step 9.4

Add $- 2492$ and $21$ .

$- 2471$