Calculus Examples

$lim_{x \to - b} {(x + b)}^{7} + \frac{{(x + b)}^{10}}{4 (x + b)}$

Step 1

Cancel the common factor of ${(x + b)}^{10}$ and $x + b$ .

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Step 1.1

Factor $x + b$ out of ${(x + b)}^{10}$ .

$lim_{x \to - b} {(x + b)}^{7} + \frac{(x + b) {(x + b)}^{9}}{4 (x + b)}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $x + b$ out of $4 (x + b)$ .

$lim_{x \to - b} {(x + b)}^{7} + \frac{(x + b) {(x + b)}^{9}}{(x + b) \cdot 4}$

Step 1.2.2

Cancel the common factor.

$lim_{x \to - b} {(x + b)}^{7} + \frac{(x + b) {(x + b)}^{9}}{(x + b) \cdot 4}$

Step 1.2.3

Rewrite the expression.

$lim_{x \to - b} {(x + b)}^{7} + \frac{{(x + b)}^{9}}{4}$

Step 2

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

$lim_{x \to - b} {(x + b)}^{7} + lim_{x \to - b} \frac{{(x + b)}^{9}}{4}$

Step 3

Move the exponent $7$ from ${(x + b)}^{7}$ outside the limit using the Limits Power Rule.

${(lim_{x \to - b} x + b)}^{7} + lim_{x \to - b} \frac{{(x + b)}^{9}}{4}$

Step 4

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

${(lim_{x \to - b} x + lim_{x \to - b} b)}^{7} + lim_{x \to - b} \frac{{(x + b)}^{9}}{4}$

Step 5

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

${(lim_{x \to - b} x + b)}^{7} + lim_{x \to - b} \frac{{(x + b)}^{9}}{4}$

Step 6

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

${(lim_{x \to - b} x + b)}^{7} + \frac{1}{4} lim_{x \to - b} {(x + b)}^{9}$

Step 7

Move the exponent $9$ from ${(x + b)}^{9}$ outside the limit using the Limits Power Rule.

${(lim_{x \to - b} x + b)}^{7} + \frac{1}{4} {(lim_{x \to - b} x + b)}^{9}$

Step 8

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

${(lim_{x \to - b} x + b)}^{7} + \frac{1}{4} {(lim_{x \to - b} x + lim_{x \to - b} b)}^{9}$

Step 9

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

${(lim_{x \to - b} x + b)}^{7} + \frac{1}{4} {(lim_{x \to - b} x + b)}^{9}$

Step 10

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

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Step 10.1

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

${(- b + b)}^{7} + \frac{1}{4} {(lim_{x \to - b} x + b)}^{9}$

Step 10.2

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

${(- b + b)}^{7} + \frac{1}{4} {(- b + b)}^{9}$

Step 11

Simplify the answer.

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Step 11.1

Combine the opposite terms in ${(- b + b)}^{7} + \frac{1}{4} {(- b + b)}^{9}$ .

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Step 11.1.1

Add $- b$ and $b$ .

$0^{7} + \frac{1}{4} {(- b + b)}^{9}$

Step 11.1.2

Add $- b$ and $b$ .

$0^{7} + \frac{1}{4} \cdot 0^{9}$

Step 11.2

Simplify each term.

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Step 11.2.1

Raising $0$ to any positive power yields $0$ .

$0 + \frac{1}{4} \cdot 0^{9}$

Step 11.2.2

Raising $0$ to any positive power yields $0$ .

$0 + \frac{1}{4} \cdot 0$

Step 11.2.3

Multiply $\frac{1}{4}$ by $0$ .

$0 + 0$

Step 11.3

Add $0$ and $0$ .

$0$