Calculus Examples

$lim_{x \to \frac{1}{5}} \frac{- 2 - 15 x}{50 x^{2} + 4}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{1}{5}$ .

$\frac{lim_{x \to \frac{1}{5}} - 2 - 15 x}{lim_{x \to \frac{1}{5}} 50 x^{2} + 4}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{5}$ .

$\frac{- lim_{x \to \frac{1}{5}} 2 - lim_{x \to \frac{1}{5}} 15 x}{lim_{x \to \frac{1}{5}} 50 x^{2} + 4}$

Step 3

Evaluate the limit of $2$ which is constant as $x$ approaches $\frac{1}{5}$ .

$\frac{- 1 \cdot 2 - lim_{x \to \frac{1}{5}} 15 x}{lim_{x \to \frac{1}{5}} 50 x^{2} + 4}$

Step 4

Move the term $15$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 2 - 15 lim_{x \to \frac{1}{5}} x}{lim_{x \to \frac{1}{5}} 50 x^{2} + 4}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{5}$ .

$\frac{- 2 - 15 lim_{x \to \frac{1}{5}} x}{lim_{x \to \frac{1}{5}} 50 x^{2} + lim_{x \to \frac{1}{5}} 4}$

Step 6

Move the term $50$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 2 - 15 lim_{x \to \frac{1}{5}} x}{50 lim_{x \to \frac{1}{5}} x^{2} + lim_{x \to \frac{1}{5}} 4}$

Step 7

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

$\frac{- 2 - 15 lim_{x \to \frac{1}{5}} x}{50 {(lim_{x \to \frac{1}{5}} x)}^{2} + lim_{x \to \frac{1}{5}} 4}$

Step 8

Evaluate the limit of $4$ which is constant as $x$ approaches $\frac{1}{5}$ .

$\frac{- 2 - 15 lim_{x \to \frac{1}{5}} x}{50 {(lim_{x \to \frac{1}{5}} x)}^{2} + 4}$

Step 9

Evaluate the limits by plugging in $\frac{1}{5}$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $\frac{1}{5}$ for $x$ .

$\frac{- 2 - 15 (\frac{1}{5})}{50 {(lim_{x \to \frac{1}{5}} x)}^{2} + 4}$

Step 9.2

Evaluate the limit of $x$ by plugging in $\frac{1}{5}$ for $x$ .

$\frac{- 2 - 15 (\frac{1}{5})}{50 {(\frac{1}{5})}^{2} + 4}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Cancel the common factor of $5$ .

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Step 10.1.1.1

Factor $5$ out of $- 15$ .

$\frac{- 2 + 5 (- 3) \frac{1}{5}}{50 {(\frac{1}{5})}^{2} + 4}$

Step 10.1.1.2

Cancel the common factor.

$\frac{- 2 + 5 \cdot - 3 \frac{1}{5}}{50 {(\frac{1}{5})}^{2} + 4}$

Step 10.1.1.3

Rewrite the expression.

$\frac{- 2 - 3}{50 {(\frac{1}{5})}^{2} + 4}$

Step 10.1.2

Subtract $3$ from $- 2$ .

$\frac{- 5}{50 {(\frac{1}{5})}^{2} + 4}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Apply the product rule to $\frac{1}{5}$ .

$\frac{- 5}{50 \frac{1^{2}}{5^{2}} + 4}$

Step 10.2.2

One to any power is one.

$\frac{- 5}{50 \frac{1}{5^{2}} + 4}$

Step 10.2.3

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

$\frac{- 5}{50 (\frac{1}{25}) + 4}$

Step 10.2.4

Cancel the common factor of $25$ .

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Step 10.2.4.1

Factor $25$ out of $50$ .

$\frac{- 5}{25 (2) \frac{1}{25} + 4}$

Step 10.2.4.2

Cancel the common factor.

$\frac{- 5}{25 \cdot 2 \frac{1}{25} + 4}$

Step 10.2.4.3

Rewrite the expression.

$\frac{- 5}{2 + 4}$

Step 10.2.5

Add $2$ and $4$ .

$\frac{- 5}{6}$

Step 10.3

Move the negative in front of the fraction.

$- \frac{5}{6}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{5}{6}$

Decimal Form:

$- 0.8\overline{3}$