Calculus Examples

$lim_{x \to 8} \frac{2 x - 5 + 4 x^{2}}{3 - 5 x + x^{2}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} 2 x - 5 + 4 x^{2}}{lim_{x \to 8} 3 - 5 x + x^{2}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} 2 x - lim_{x \to 8} 5 + lim_{x \to 8} 4 x^{2}}{lim_{x \to 8} 3 - 5 x + x^{2}}$

Step 3

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 lim_{x \to 8} x - lim_{x \to 8} 5 + lim_{x \to 8} 4 x^{2}}{lim_{x \to 8} 3 - 5 x + x^{2}}$

Step 4

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

$\frac{2 lim_{x \to 8} x - 1 \cdot 5 + lim_{x \to 8} 4 x^{2}}{lim_{x \to 8} 3 - 5 x + x^{2}}$

Step 5

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 lim_{x \to 8} x - 1 \cdot 5 + 4 lim_{x \to 8} x^{2}}{lim_{x \to 8} 3 - 5 x + x^{2}}$

Step 6

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

$\frac{2 lim_{x \to 8} x - 1 \cdot 5 + 4 {(lim_{x \to 8} x)}^{2}}{lim_{x \to 8} 3 - 5 x + x^{2}}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{2 lim_{x \to 8} x - 1 \cdot 5 + 4 {(lim_{x \to 8} x)}^{2}}{lim_{x \to 8} 3 - lim_{x \to 8} 5 x + lim_{x \to 8} x^{2}}$

Step 8

Evaluate the limit of $3$ which is constant as $x$ approaches $8$ .

$\frac{2 lim_{x \to 8} x - 1 \cdot 5 + 4 {(lim_{x \to 8} x)}^{2}}{3 - lim_{x \to 8} 5 x + lim_{x \to 8} x^{2}}$

Step 9

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 lim_{x \to 8} x - 1 \cdot 5 + 4 {(lim_{x \to 8} x)}^{2}}{3 - 5 lim_{x \to 8} x + lim_{x \to 8} x^{2}}$

Step 10

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

$\frac{2 lim_{x \to 8} x - 1 \cdot 5 + 4 {(lim_{x \to 8} x)}^{2}}{3 - 5 lim_{x \to 8} x + {(lim_{x \to 8} x)}^{2}}$

Step 11

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 11.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{2 \cdot 8 - 1 \cdot 5 + 4 {(lim_{x \to 8} x)}^{2}}{3 - 5 lim_{x \to 8} x + {(lim_{x \to 8} x)}^{2}}$

Step 11.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{2 \cdot 8 - 1 \cdot 5 + 4 \cdot 8^{2}}{3 - 5 lim_{x \to 8} x + {(lim_{x \to 8} x)}^{2}}$

Step 11.3

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{2 \cdot 8 - 1 \cdot 5 + 4 \cdot 8^{2}}{3 - 5 \cdot 8 + {(lim_{x \to 8} x)}^{2}}$

Step 11.4

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{2 \cdot 8 - 1 \cdot 5 + 4 \cdot 8^{2}}{3 - 5 \cdot 8 + 8^{2}}$

Step 12

Simplify the answer.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

Multiply $2$ by $8$ .

$\frac{16 - 1 \cdot 5 + 4 \cdot 8^{2}}{3 - 5 \cdot 8 + 8^{2}}$

Step 12.1.2

Multiply $- 1$ by $5$ .

$\frac{16 - 5 + 4 \cdot 8^{2}}{3 - 5 \cdot 8 + 8^{2}}$

Step 12.1.3

Raise $8$ to the power of $2$ .

$\frac{16 - 5 + 4 \cdot 64}{3 - 5 \cdot 8 + 8^{2}}$

Step 12.1.4

Multiply $4$ by $64$ .

$\frac{16 - 5 + 256}{3 - 5 \cdot 8 + 8^{2}}$

Step 12.1.5

Subtract $5$ from $16$ .

$\frac{11 + 256}{3 - 5 \cdot 8 + 8^{2}}$

Step 12.1.6

Add $11$ and $256$ .

$\frac{267}{3 - 5 \cdot 8 + 8^{2}}$

Step 12.2

Simplify the denominator.

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Step 12.2.1

Multiply $- 5$ by $8$ .

$\frac{267}{3 - 40 + 8^{2}}$

Step 12.2.2

Raise $8$ to the power of $2$ .

$\frac{267}{3 - 40 + 64}$

Step 12.2.3

Subtract $40$ from $3$ .

$\frac{267}{- 37 + 64}$

Step 12.2.4

Add $- 37$ and $64$ .

$\frac{267}{27}$

Step 12.3

Cancel the common factor of $267$ and $27$ .

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Step 12.3.1

Factor $3$ out of $267$ .

$\frac{3 (89)}{27}$

Step 12.3.2

Cancel the common factors.

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Step 12.3.2.1

Factor $3$ out of $27$ .

$\frac{3 \cdot 89}{3 \cdot 9}$

Step 12.3.2.2

Cancel the common factor.

$\frac{3 \cdot 89}{3 \cdot 9}$

Step 12.3.2.3

Rewrite the expression.

$\frac{89}{9}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{89}{9}$

Decimal Form:

$9.\overline{8}$