Calculus Examples

$lim_{x \to 7} \frac{1}{x^{2} (x + 7)}$

Step 1

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

$\frac{lim_{x \to 7} 1}{lim_{x \to 7} x^{2} (x + 7)}$

Step 2

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

$\frac{1}{lim_{x \to 7} x^{2} (x + 7)}$

Step 3

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

$\frac{1}{lim_{x \to 7} x^{2} \cdot lim_{x \to 7} x + 7}$

Step 4

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

$\frac{1}{{(lim_{x \to 7} x)}^{2} \cdot lim_{x \to 7} x + 7}$

Step 5

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

$\frac{1}{{(lim_{x \to 7} x)}^{2} \cdot (lim_{x \to 7} x + lim_{x \to 7} 7)}$

Step 6

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

$\frac{1}{{(lim_{x \to 7} x)}^{2} \cdot (lim_{x \to 7} x + 7)}$

Step 7

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

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Step 7.1

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

$\frac{1}{7^{2} \cdot (lim_{x \to 7} x + 7)}$

Step 7.2

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

$\frac{1}{7^{2} \cdot (7 + 7)}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the denominator.

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Step 8.1.1

Add $7$ and $7$ .

$\frac{1}{7^{2} \cdot 14}$

Step 8.1.2

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

$\frac{1}{49 \cdot 14}$

Step 8.2

Multiply $49$ by $14$ .

$\frac{1}{686}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{686}$

Decimal Form:

$0.00145772 \dots$