Calculus Examples

$lim_{x \to 5} \frac{2 x^{2} - 9 x - 5}{3 x^{3} - 8 x^{2} - 35 x}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 5} 2 x^{2} - 9 x - 5}{lim_{x \to 5} 3 x^{3} - 8 x^{2} - 35 x}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

$\frac{lim_{x \to 5} 2 x^{2} - lim_{x \to 5} 9 x - lim_{x \to 5} 5}{lim_{x \to 5} 3 x^{3} - 8 x^{2} - 35 x}$

Step 1.1.2.2

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

$\frac{2 lim_{x \to 5} x^{2} - lim_{x \to 5} 9 x - lim_{x \to 5} 5}{lim_{x \to 5} 3 x^{3} - 8 x^{2} - 35 x}$

Step 1.1.2.3

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

$\frac{2 {(lim_{x \to 5} x)}^{2} - lim_{x \to 5} 9 x - lim_{x \to 5} 5}{lim_{x \to 5} 3 x^{3} - 8 x^{2} - 35 x}$

Step 1.1.2.4

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 {(lim_{x \to 5} x)}^{2} - 9 lim_{x \to 5} x - lim_{x \to 5} 5}{lim_{x \to 5} 3 x^{3} - 8 x^{2} - 35 x}$

Step 1.1.2.5

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

$\frac{2 {(lim_{x \to 5} x)}^{2} - 9 lim_{x \to 5} x - 1 \cdot 5}{lim_{x \to 5} 3 x^{3} - 8 x^{2} - 35 x}$

Step 1.1.2.6

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

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Step 1.1.2.6.1

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

$\frac{2 \cdot 5^{2} - 9 lim_{x \to 5} x - 1 \cdot 5}{lim_{x \to 5} 3 x^{3} - 8 x^{2} - 35 x}$

Step 1.1.2.6.2

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

$\frac{2 \cdot 5^{2} - 9 \cdot 5 - 1 \cdot 5}{lim_{x \to 5} 3 x^{3} - 8 x^{2} - 35 x}$

Step 1.1.2.7

Simplify the answer.

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Step 1.1.2.7.1

Simplify each term.

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Step 1.1.2.7.1.1

Raise $5$ to the power of $2$ .

$\frac{2 \cdot 25 - 9 \cdot 5 - 1 \cdot 5}{lim_{x \to 5} 3 x^{3} - 8 x^{2} - 35 x}$

Step 1.1.2.7.1.2

Multiply $2$ by $25$ .

$\frac{50 - 9 \cdot 5 - 1 \cdot 5}{lim_{x \to 5} 3 x^{3} - 8 x^{2} - 35 x}$

Step 1.1.2.7.1.3

Multiply $- 9$ by $5$ .

$\frac{50 - 45 - 1 \cdot 5}{lim_{x \to 5} 3 x^{3} - 8 x^{2} - 35 x}$

Step 1.1.2.7.1.4

Multiply $- 1$ by $5$ .

$\frac{50 - 45 - 5}{lim_{x \to 5} 3 x^{3} - 8 x^{2} - 35 x}$

Step 1.1.2.7.2

Subtract $45$ from $50$ .

$\frac{5 - 5}{lim_{x \to 5} 3 x^{3} - 8 x^{2} - 35 x}$

Step 1.1.2.7.3

Subtract $5$ from $5$ .

$\frac{0}{lim_{x \to 5} 3 x^{3} - 8 x^{2} - 35 x}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

$\frac{0}{lim_{x \to 5} 3 x^{3} - lim_{x \to 5} 8 x^{2} - lim_{x \to 5} 35 x}$

Step 1.1.3.2

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{3 lim_{x \to 5} x^{3} - lim_{x \to 5} 8 x^{2} - lim_{x \to 5} 35 x}$

Step 1.1.3.3

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

$\frac{0}{3 {(lim_{x \to 5} x)}^{3} - lim_{x \to 5} 8 x^{2} - lim_{x \to 5} 35 x}$

Step 1.1.3.4

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{3 {(lim_{x \to 5} x)}^{3} - 8 lim_{x \to 5} x^{2} - lim_{x \to 5} 35 x}$

Step 1.1.3.5

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

$\frac{0}{3 {(lim_{x \to 5} x)}^{3} - 8 {(lim_{x \to 5} x)}^{2} - lim_{x \to 5} 35 x}$

Step 1.1.3.6

Move the term $35$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{3 {(lim_{x \to 5} x)}^{3} - 8 {(lim_{x \to 5} x)}^{2} - 35 lim_{x \to 5} x}$

Step 1.1.3.7

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

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Step 1.1.3.7.1

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

$\frac{0}{3 \cdot 5^{3} - 8 {(lim_{x \to 5} x)}^{2} - 35 lim_{x \to 5} x}$

Step 1.1.3.7.2

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

$\frac{0}{3 \cdot 5^{3} - 8 \cdot 5^{2} - 35 lim_{x \to 5} x}$

Step 1.1.3.7.3

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

$\frac{0}{3 \cdot 5^{3} - 8 \cdot 5^{2} - 35 \cdot 5}$

Step 1.1.3.8

Simplify the answer.

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Step 1.1.3.8.1

Simplify each term.

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Step 1.1.3.8.1.1

Raise $5$ to the power of $3$ .

$\frac{0}{3 \cdot 125 - 8 \cdot 5^{2} - 35 \cdot 5}$

Step 1.1.3.8.1.2

Multiply $3$ by $125$ .

$\frac{0}{375 - 8 \cdot 5^{2} - 35 \cdot 5}$

Step 1.1.3.8.1.3

Raise $5$ to the power of $2$ .

$\frac{0}{375 - 8 \cdot 25 - 35 \cdot 5}$

Step 1.1.3.8.1.4

Multiply $- 8$ by $25$ .

$\frac{0}{375 - 200 - 35 \cdot 5}$

Step 1.1.3.8.1.5

Multiply $- 35$ by $5$ .

$\frac{0}{375 - 200 - 175}$

Step 1.1.3.8.2

Subtract $200$ from $375$ .

$\frac{0}{175 - 175}$

Step 1.1.3.8.3

Subtract $175$ from $175$ .

$\frac{0}{0}$

Step 1.1.3.8.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.3.9

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 5} \frac{2 x^{2} - 9 x - 5}{3 x^{3} - 8 x^{2} - 35 x} = lim_{x \to 5} \frac{\frac{d}{d x} [2 x^{2} - 9 x - 5]}{\frac{d}{d x} [3 x^{3} - 8 x^{2} - 35 x]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 5} \frac{\frac{d}{d x} [2 x^{2} - 9 x - 5]}{\frac{d}{d x} [3 x^{3} - 8 x^{2} - 35 x]}$

Step 1.3.2

By the Sum Rule, the derivative of $2 x^{2} - 9 x - 5$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [- 5]$ .

$lim_{x \to 5} \frac{\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [- 5]}{\frac{d}{d x} [3 x^{3} - 8 x^{2} - 35 x]}$

Step 1.3.3

Evaluate $\frac{d}{d x} [2 x^{2}]$ .

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Step 1.3.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{2}$ with respect to $x$ is $2 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to 5} \frac{2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [- 5]}{\frac{d}{d x} [3 x^{3} - 8 x^{2} - 35 x]}$

Step 1.3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 5} \frac{2 (2 x) + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [- 5]}{\frac{d}{d x} [3 x^{3} - 8 x^{2} - 35 x]}$

Step 1.3.3.3

Multiply $2$ by $2$ .

$lim_{x \to 5} \frac{4 x + \frac{d}{d x} [- 9 x] + \frac{d}{d x} [- 5]}{\frac{d}{d x} [3 x^{3} - 8 x^{2} - 35 x]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- 9 x]$ .

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Step 1.3.4.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x$ with respect to $x$ is $- 9 \frac{d}{d x} [x]$ .

$lim_{x \to 5} \frac{4 x - 9 \frac{d}{d x} [x] + \frac{d}{d x} [- 5]}{\frac{d}{d x} [3 x^{3} - 8 x^{2} - 35 x]}$

Step 1.3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 5} \frac{4 x - 9 \cdot 1 + \frac{d}{d x} [- 5]}{\frac{d}{d x} [3 x^{3} - 8 x^{2} - 35 x]}$

Step 1.3.4.3

Multiply $- 9$ by $1$ .

$lim_{x \to 5} \frac{4 x - 9 + \frac{d}{d x} [- 5]}{\frac{d}{d x} [3 x^{3} - 8 x^{2} - 35 x]}$

Step 1.3.5

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$lim_{x \to 5} \frac{4 x - 9 + 0}{\frac{d}{d x} [3 x^{3} - 8 x^{2} - 35 x]}$

Step 1.3.6

Add $4 x - 9$ and $0$ .

$lim_{x \to 5} \frac{4 x - 9}{\frac{d}{d x} [3 x^{3} - 8 x^{2} - 35 x]}$

Step 1.3.7

By the Sum Rule, the derivative of $3 x^{3} - 8 x^{2} - 35 x$ with respect to $x$ is $\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [- 8 x^{2}] + \frac{d}{d x} [- 35 x]$ .

$lim_{x \to 5} \frac{4 x - 9}{\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [- 8 x^{2}] + \frac{d}{d x} [- 35 x]}$

Step 1.3.8

Evaluate $\frac{d}{d x} [3 x^{3}]$ .

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Step 1.3.8.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{3}$ with respect to $x$ is $3 \frac{d}{d x} [x^{3}]$ .

$lim_{x \to 5} \frac{4 x - 9}{3 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 8 x^{2}] + \frac{d}{d x} [- 35 x]}$

Step 1.3.8.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$lim_{x \to 5} \frac{4 x - 9}{3 (3 x^{2}) + \frac{d}{d x} [- 8 x^{2}] + \frac{d}{d x} [- 35 x]}$

Step 1.3.8.3

Multiply $3$ by $3$ .

$lim_{x \to 5} \frac{4 x - 9}{9 x^{2} + \frac{d}{d x} [- 8 x^{2}] + \frac{d}{d x} [- 35 x]}$

Step 1.3.9

Evaluate $\frac{d}{d x} [- 8 x^{2}]$ .

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Step 1.3.9.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{2}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to 5} \frac{4 x - 9}{9 x^{2} - 8 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 35 x]}$

Step 1.3.9.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 5} \frac{4 x - 9}{9 x^{2} - 8 (2 x) + \frac{d}{d x} [- 35 x]}$

Step 1.3.9.3

Multiply $2$ by $- 8$ .

$lim_{x \to 5} \frac{4 x - 9}{9 x^{2} - 16 x + \frac{d}{d x} [- 35 x]}$

Step 1.3.10

Evaluate $\frac{d}{d x} [- 35 x]$ .

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Step 1.3.10.1

Since $- 35$ is constant with respect to $x$ , the derivative of $- 35 x$ with respect to $x$ is $- 35 \frac{d}{d x} [x]$ .

$lim_{x \to 5} \frac{4 x - 9}{9 x^{2} - 16 x - 35 \frac{d}{d x} [x]}$

Step 1.3.10.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 5} \frac{4 x - 9}{9 x^{2} - 16 x - 35 \cdot 1}$

Step 1.3.10.3

Multiply $- 35$ by $1$ .

$lim_{x \to 5} \frac{4 x - 9}{9 x^{2} - 16 x - 35}$

Step 2

Evaluate the limit.

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Step 2.1

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

$\frac{lim_{x \to 5} 4 x - 9}{lim_{x \to 5} 9 x^{2} - 16 x - 35}$

Step 2.2

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

$\frac{lim_{x \to 5} 4 x - lim_{x \to 5} 9}{lim_{x \to 5} 9 x^{2} - 16 x - 35}$

Step 2.3

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

$\frac{4 lim_{x \to 5} x - lim_{x \to 5} 9}{lim_{x \to 5} 9 x^{2} - 16 x - 35}$

Step 2.4

Evaluate the limit of $9$ which is constant as $x$ approaches $5$ .

$\frac{4 lim_{x \to 5} x - 1 \cdot 9}{lim_{x \to 5} 9 x^{2} - 16 x - 35}$

Step 2.5

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

$\frac{4 lim_{x \to 5} x - 1 \cdot 9}{lim_{x \to 5} 9 x^{2} - lim_{x \to 5} 16 x - lim_{x \to 5} 35}$

Step 2.6

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 lim_{x \to 5} x - 1 \cdot 9}{9 lim_{x \to 5} x^{2} - lim_{x \to 5} 16 x - lim_{x \to 5} 35}$

Step 2.7

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

$\frac{4 lim_{x \to 5} x - 1 \cdot 9}{9 {(lim_{x \to 5} x)}^{2} - lim_{x \to 5} 16 x - lim_{x \to 5} 35}$

Step 2.8

Move the term $16$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 lim_{x \to 5} x - 1 \cdot 9}{9 {(lim_{x \to 5} x)}^{2} - 16 lim_{x \to 5} x - lim_{x \to 5} 35}$

Step 2.9

Evaluate the limit of $35$ which is constant as $x$ approaches $5$ .

$\frac{4 lim_{x \to 5} x - 1 \cdot 9}{9 {(lim_{x \to 5} x)}^{2} - 16 lim_{x \to 5} x - 1 \cdot 35}$

Step 3

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

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Step 3.1

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

$\frac{4 \cdot 5 - 1 \cdot 9}{9 {(lim_{x \to 5} x)}^{2} - 16 lim_{x \to 5} x - 1 \cdot 35}$

Step 3.2

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

$\frac{4 \cdot 5 - 1 \cdot 9}{9 \cdot 5^{2} - 16 lim_{x \to 5} x - 1 \cdot 35}$

Step 3.3

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

$\frac{4 \cdot 5 - 1 \cdot 9}{9 \cdot 5^{2} - 16 \cdot 5 - 1 \cdot 35}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $4$ by $5$ .

$\frac{20 - 1 \cdot 9}{9 \cdot 5^{2} - 16 \cdot 5 - 1 \cdot 35}$

Step 4.1.2

Multiply $- 1$ by $9$ .

$\frac{20 - 9}{9 \cdot 5^{2} - 16 \cdot 5 - 1 \cdot 35}$

Step 4.1.3

Subtract $9$ from $20$ .

$\frac{11}{9 \cdot 5^{2} - 16 \cdot 5 - 1 \cdot 35}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Raise $5$ to the power of $2$ .

$\frac{11}{9 \cdot 25 - 16 \cdot 5 - 1 \cdot 35}$

Step 4.2.2

Multiply $9$ by $25$ .

$\frac{11}{225 - 16 \cdot 5 - 1 \cdot 35}$

Step 4.2.3

Multiply $- 16$ by $5$ .

$\frac{11}{225 - 80 - 1 \cdot 35}$

Step 4.2.4

Multiply $- 1$ by $35$ .

$\frac{11}{225 - 80 - 35}$

Step 4.2.5

Subtract $80$ from $225$ .

$\frac{11}{145 - 35}$

Step 4.2.6

Subtract $35$ from $145$ .

$\frac{11}{110}$

Step 4.3

Cancel the common factor of $11$ and $110$ .

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Step 4.3.1

Factor $11$ out of $11$ .

$\frac{11 (1)}{110}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $11$ out of $110$ .

$\frac{11 \cdot 1}{11 \cdot 10}$

Step 4.3.2.2

Cancel the common factor.

$\frac{11 \cdot 1}{11 \cdot 10}$

Step 4.3.2.3

Rewrite the expression.

$\frac{1}{10}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{10}$

Decimal Form:

$0.1$