Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Move the exponent $10$ from ${(x + b)}^{10}$ outside the limit using the Limits Power Rule.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Factor ${(- b + b)}^{7}$ out of ${(- b + b)}^{7} + {(- b + b)}^{10}$ .

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Step 9.1.1.1

Multiply by $1$ .

$\frac{{(- b + b)}^{7} \cdot 1 + {(- b + b)}^{10}}{4 (x + b)}$

Step 9.1.1.2

Factor ${(- b + b)}^{7}$ out of ${(- b + b)}^{10}$ .

$\frac{{(- b + b)}^{7} \cdot 1 + {(- b + b)}^{7} {(- b + b)}^{3}}{4 (x + b)}$

Step 9.1.1.3

Factor ${(- b + b)}^{7}$ out of ${(- b + b)}^{7} \cdot 1 + {(- b + b)}^{7} {(- b + b)}^{3}$ .

$\frac{{(- b + b)}^{7} (1 + {(- b + b)}^{3})}{4 (x + b)}$

Step 9.1.2

Add $- b$ and $b$ .

$\frac{0^{7} (1 + {(- b + b)}^{3})}{4 (x + b)}$

Step 9.1.3

Add $- b$ and $b$ .

$\frac{0^{7} (1 + 0^{3})}{4 (x + b)}$

Step 9.1.4

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

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

Step 9.1.5

Add $1$ and $0$ .

$\frac{0^{7} \cdot 1}{4 (x + b)}$

Step 9.1.6

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

$\frac{0 \cdot 1}{4 (x + b)}$

Step 9.2

Multiply $0$ by $1$ .

$\frac{0}{4 (x + b)}$

Step 9.3

Cancel the common factor of $0$ and $4$ .

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Step 9.3.1

Factor $4$ out of $0$ .

$\frac{4 (0)}{4 (x + b)}$

Step 9.3.2

Cancel the common factors.

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Step 9.3.2.1

Cancel the common factor.

$\frac{4 \cdot 0}{4 (x + b)}$

Step 9.3.2.2

Rewrite the expression.

$\frac{0}{x + b}$

Step 9.4

Divide $0$ by $x + b$ .

$0$