Calculus Examples

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

Step 1

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

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

Step 2

Move the exponent $2 - 25$ from $x^{2 - 25}$ outside the limit using the Limits Power Rule.

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

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 $x$ .

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

Step 3.2

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

$\frac{5}{5^{2 - 25}}$

Step 4

Simplify the answer.

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Step 4.1

Move $5^{2 - 25}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$5 \cdot 5^{- (2 - 25)}$

Step 4.2

Multiply $5$ by $5^{- (2 - 25)}$ by adding the exponents.

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Step 4.2.1

Multiply $5$ by $5^{- (2 - 25)}$ .

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Step 4.2.1.1

Raise $5$ to the power of $1$ .

$5^{1} \cdot 5^{- (2 - 25)}$

Step 4.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5^{1 - (2 - 25)}$

Step 4.2.2

Subtract $25$ from $2$ .

$5^{1 - - 23}$

Step 4.2.3

Multiply $- 1$ by $- 23$ .

$5^{1 + 23}$

Step 4.2.4

Add $1$ and $23$ .

$5^{24}$

Step 4.3

Raise $5$ to the power of $24$ .

$59604644775390600$

Step 5

The result can be shown in multiple forms.

Exact Form:

$59604644775390600$

Decimal Form:

$5.96046447 \cdot 10^{16}$