Calculus Examples

$lim_{x \to 5} \frac{{(x - 5)}^{2}}{{(x - 3)}^{2} {(x - 8)}^{2}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $5$ .

$\frac{lim_{x \to 5} {(x - 5)}^{2}}{lim_{x \to 5} {(x - 3)}^{2} {(x - 8)}^{2}}$

Step 2

Move the exponent $2$ from ${(x - 5)}^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 5} x - 5)}^{2}}{lim_{x \to 5} {(x - 3)}^{2} {(x - 8)}^{2}}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{{(lim_{x \to 5} x - lim_{x \to 5} 5)}^{2}}{lim_{x \to 5} {(x - 3)}^{2} {(x - 8)}^{2}}$

Step 4

Evaluate the limit of $5$ which is constant as $x$ approaches $5$ .

$\frac{{(lim_{x \to 5} x - 1 \cdot 5)}^{2}}{lim_{x \to 5} {(x - 3)}^{2} {(x - 8)}^{2}}$

Step 5

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{{(lim_{x \to 5} x - 1 \cdot 5)}^{2}}{lim_{x \to 5} {(x - 3)}^{2} \cdot lim_{x \to 5} {(x - 8)}^{2}}$

Step 6

Move the exponent $2$ from ${(x - 3)}^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 5} x - 1 \cdot 5)}^{2}}{{(lim_{x \to 5} x - 3)}^{2} \cdot lim_{x \to 5} {(x - 8)}^{2}}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{{(lim_{x \to 5} x - 1 \cdot 5)}^{2}}{{(lim_{x \to 5} x - lim_{x \to 5} 3)}^{2} \cdot lim_{x \to 5} {(x - 8)}^{2}}$

Step 8

Evaluate the limit of $3$ which is constant as $x$ approaches $5$ .

$\frac{{(lim_{x \to 5} x - 1 \cdot 5)}^{2}}{{(lim_{x \to 5} x - 1 \cdot 3)}^{2} \cdot lim_{x \to 5} {(x - 8)}^{2}}$

Step 9

Move the exponent $2$ from ${(x - 8)}^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 5} x - 1 \cdot 5)}^{2}}{{(lim_{x \to 5} x - 1 \cdot 3)}^{2} \cdot {(lim_{x \to 5} x - 8)}^{2}}$

Step 10

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{{(lim_{x \to 5} x - 1 \cdot 5)}^{2}}{{(lim_{x \to 5} x - 1 \cdot 3)}^{2} \cdot {(lim_{x \to 5} x - lim_{x \to 5} 8)}^{2}}$

Step 11

Evaluate the limit of $8$ which is constant as $x$ approaches $5$ .

$\frac{{(lim_{x \to 5} x - 1 \cdot 5)}^{2}}{{(lim_{x \to 5} x - 1 \cdot 3)}^{2} \cdot {(lim_{x \to 5} x - 1 \cdot 8)}^{2}}$

Step 12

Evaluate the limits by plugging in $5$ for all occurrences of $x$ .

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Step 12.1

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{{(5 - 1 \cdot 5)}^{2}}{{(lim_{x \to 5} x - 1 \cdot 3)}^{2} \cdot {(lim_{x \to 5} x - 1 \cdot 8)}^{2}}$

Step 12.2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{{(5 - 1 \cdot 5)}^{2}}{{(5 - 1 \cdot 3)}^{2} \cdot {(lim_{x \to 5} x - 1 \cdot 8)}^{2}}$

Step 12.3

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{{(5 - 1 \cdot 5)}^{2}}{{(5 - 1 \cdot 3)}^{2} \cdot {(5 - 1 \cdot 8)}^{2}}$

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Multiply $- 1$ by $5$ .

$\frac{{(5 - 5)}^{2}}{{(5 - 1 \cdot 3)}^{2} \cdot {(5 - 1 \cdot 8)}^{2}}$

Step 13.1.2

Subtract $5$ from $5$ .

$\frac{0^{2}}{{(5 - 1 \cdot 3)}^{2} \cdot {(5 - 1 \cdot 8)}^{2}}$

Step 13.1.3

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

$\frac{0}{{(5 - 1 \cdot 3)}^{2} \cdot {(5 - 1 \cdot 8)}^{2}}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Multiply $- 1$ by $3$ .

$\frac{0}{{(5 - 3)}^{2} {(5 - 1 \cdot 8)}^{2}}$

Step 13.2.2

Subtract $3$ from $5$ .

$\frac{0}{2^{2} {(5 - 1 \cdot 8)}^{2}}$

Step 13.2.3

Multiply $- 1$ by $8$ .

$\frac{0}{2^{2} {(5 - 8)}^{2}}$

Step 13.2.4

Subtract $8$ from $5$ .

$\frac{0}{2^{2} {(- 3)}^{2}}$

Step 13.2.5

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

$\frac{0}{4 {(- 3)}^{2}}$

Step 13.2.6

Raise $- 3$ to the power of $2$ .

$\frac{0}{4 \cdot 9}$

Step 13.3

Multiply $4$ by $9$ .

$\frac{0}{36}$

Step 13.4

Divide $0$ by $36$ .

$0$