Calculus Examples

$lim_{x \to 5} (\frac{x}{x^{2} - 25}) - \frac{5}{x^{2} - 25}$

Step 1

Combine fractions using a common denominator.

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Step 1.1

Combine the numerators over the common denominator.

$lim_{x \to 5} \frac{x - 5}{x^{2} - 25}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Rewrite $25$ as $5^{2}$ .

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

Step 1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$lim_{x \to 5} \frac{x - 5}{(x + 5) (x - 5)}$

Step 1.3

Cancel the common factor of $x - 5$ .

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Step 1.3.1

Cancel the common factor.

$lim_{x \to 5} \frac{x - 5}{(x + 5) (x - 5)}$

Step 1.3.2

Rewrite the expression.

$lim_{x \to 5} \frac{1}{x + 5}$

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 $5$ .

$\frac{lim_{x \to 5} 1}{lim_{x \to 5} x + 5}$

Step 2.2

Evaluate the limit of $1$ which is constant as $x$ approaches $5$ .

$\frac{1}{lim_{x \to 5} x + 5}$

Step 2.3

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

$\frac{1}{lim_{x \to 5} x + lim_{x \to 5} 5}$

Step 2.4

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

$\frac{1}{lim_{x \to 5} x + 5}$

Step 3

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

$\frac{1}{5 + 5}$

Step 4

Add $5$ and $5$ .

$\frac{1}{10}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{10}$

Decimal Form:

$0.1$