Calculus Examples

$\frac{lim_{x \to i n f} 7 x^{4} - 6 x^{2}}{5 x^{5} + 3}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $i n f$ .

$\frac{lim_{x \to i n f} 7 x^{4} - lim_{x \to i n f} 6 x^{2}}{5 x^{5} + 3}$

Step 2

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

$\frac{7 lim_{x \to i n f} x^{4} - lim_{x \to i n f} 6 x^{2}}{5 x^{5} + 3}$

Step 3

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$\frac{7 {(lim_{x \to i n f} x)}^{4} - lim_{x \to i n f} 6 x^{2}}{5 x^{5} + 3}$

Step 4

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{7 {(lim_{x \to i n f} x)}^{4} - 6 lim_{x \to i n f} x^{2}}{5 x^{5} + 3}$

Step 5

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

$\frac{7 {(lim_{x \to i n f} x)}^{4} - 6 {(lim_{x \to i n f} x)}^{2}}{5 x^{5} + 3}$

Step 6

Evaluate the limits by plugging in $i n f$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $i n f$ for $x$ .

$\frac{7 {(i n f)}^{4} - 6 {(lim_{x \to i n f} x)}^{2}}{5 x^{5} + 3}$

Step 6.2

Evaluate the limit of $x$ by plugging in $i n f$ for $x$ .

$\frac{7 {(i n f)}^{4} - 6 {(i n f)}^{2}}{5 x^{5} + 3}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Factor ${(i n f)}^{2}$ out of $7 {(i n f)}^{4} - 6 {(i n f)}^{2}$ .

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Step 7.1.1.1

Factor ${(i n f)}^{2}$ out of $7 {(i n f)}^{4}$ .

$\frac{{(i n f)}^{2} (7 {(i n f)}^{2}) - 6 {(i n f)}^{2}}{5 x^{5} + 3}$

Step 7.1.1.2

Factor ${(i n f)}^{2}$ out of $- 6 {(i n f)}^{2}$ .

$\frac{{(i n f)}^{2} (7 {(i n f)}^{2}) + {(i n f)}^{2} \cdot - 6}{5 x^{5} + 3}$

Step 7.1.1.3

Factor ${(i n f)}^{2}$ out of ${(i n f)}^{2} (7 {(i n f)}^{2}) + {(i n f)}^{2} \cdot - 6$ .

$\frac{{(i n f)}^{2} (7 {(i n f)}^{2} - 6)}{5 x^{5} + 3}$

Step 7.1.2

Apply the product rule to $i n f$ .

$\frac{{(i n)}^{2} f^{2} (7 {(i n f)}^{2} - 6)}{5 x^{5} + 3}$

Step 7.1.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 7.1.3.1

Apply the product rule to $i n f$ .

$\frac{{(i n)}^{2} f^{2} (7 ({(i n)}^{2} f^{2}) - 6)}{5 x^{5} + 3}$

Step 7.1.3.2

Apply the product rule to $i n$ .

$\frac{{(i n)}^{2} f^{2} (7 (i^{2} n^{2} f^{2}) - 6)}{5 x^{5} + 3}$

Step 7.1.4

Rewrite $i^{2}$ as $- 1$ .

$\frac{{(i n)}^{2} f^{2} (7 (- 1 n^{2} f^{2}) - 6)}{5 x^{5} + 3}$

Step 7.1.5

Rewrite $- 1 n^{2}$ as $- n^{2}$ .

$\frac{{(i n)}^{2} f^{2} (7 (- n^{2} f^{2}) - 6)}{5 x^{5} + 3}$

Step 7.1.6

Multiply $- 1$ by $7$ .

$\frac{{(i n)}^{2} f^{2} (- 7 (n^{2} f^{2}) - 6)}{5 x^{5} + 3}$

Step 7.1.7

Apply the product rule to $i n$ .

$\frac{i^{2} n^{2} f^{2} (- 7 n^{2} f^{2} - 6)}{5 x^{5} + 3}$

Step 7.1.8

Rewrite $i^{2}$ as $- 1$ .

$\frac{- 1 n^{2} f^{2} (- 7 n^{2} f^{2} - 6)}{5 x^{5} + 3}$

Step 7.1.9

Rewrite $- 1 n^{2}$ as $- n^{2}$ .

$\frac{- n^{2} f^{2} (- 7 n^{2} f^{2} - 6)}{5 x^{5} + 3}$

Step 7.2

Move the negative in front of the fraction.

$- \frac{n^{2} f^{2} (- 7 n^{2} f^{2} - 6)}{5 x^{5} + 3}$

Step 7.3

Factor $- 1$ out of $- 7 n^{2} f^{2}$ .

$- \frac{n^{2} f^{2} (- (7 n^{2} f^{2}) - 6)}{5 x^{5} + 3}$

Step 7.4

Rewrite $- 6$ as $- 1 (6)$ .

$- \frac{n^{2} f^{2} (- (7 n^{2} f^{2}) - 1 (6))}{5 x^{5} + 3}$

Step 7.5

Factor $- 1$ out of $- (7 n^{2} f^{2}) - 1 (6)$ .

$- \frac{n^{2} f^{2} (- (7 n^{2} f^{2} + 6))}{5 x^{5} + 3}$

Step 7.6

Rewrite $- (7 n^{2} f^{2} + 6)$ as $- 1 (7 n^{2} f^{2} + 6)$ .

$- \frac{n^{2} f^{2} (- 1 (7 n^{2} f^{2} + 6))}{5 x^{5} + 3}$

Step 7.7

Move the negative in front of the fraction.

$- - \frac{(n^{2} f^{2}) (7 n^{2} f^{2} + 6)}{5 x^{5} + 3}$

Step 7.8

Multiply $- 1$ by $- 1$ .

$1 \frac{(n^{2} f^{2}) (7 n^{2} f^{2} + 6)}{5 x^{5} + 3}$

Step 7.9

Multiply $\frac{(n^{2} f^{2}) (7 n^{2} f^{2} + 6)}{5 x^{5} + 3}$ by $1$ .

$\frac{(n^{2} f^{2}) (7 n^{2} f^{2} + 6)}{5 x^{5} + 3}$

Step 7.10

Reorder factors in $\frac{(n^{2} f^{2}) (7 n^{2} f^{2} + 6)}{5 x^{5} + 3}$ .

$\frac{n^{2} f^{2} (7 n^{2} f^{2} + 6)}{5 x^{5} + 3}$