Calculus Examples

$lim_{t \to 25} (t + 5) t^{- \frac{1}{2}}$

Step 1

Split the limit using the Product of Limits Rule on the limit as $t$ approaches $25$ .

$lim_{t \to 25} t + 5 \cdot lim_{t \to 25} t^{- \frac{1}{2}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $25$ .

$(lim_{t \to 25} t + lim_{t \to 25} 5) \cdot lim_{t \to 25} t^{- \frac{1}{2}}$

Step 3

Evaluate the limit of $5$ which is constant as $t$ approaches $25$ .

$(lim_{t \to 25} t + 5) \cdot lim_{t \to 25} t^{- \frac{1}{2}}$

Step 4

Move the exponent $- \frac{1}{2}$ from $t^{- \frac{1}{2}}$ outside the limit using the Limits Power Rule.

$(lim_{t \to 25} t + 5) \cdot {(lim_{t \to 25} t)}^{- \frac{1}{2}}$

Step 5

Evaluate the limits by plugging in $25$ for all occurrences of $t$ .

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Step 5.1

Evaluate the limit of $t$ by plugging in $25$ for $t$ .

$(25 + 5) \cdot {(lim_{t \to 25} t)}^{- \frac{1}{2}}$

Step 5.2

Evaluate the limit of $t$ by plugging in $25$ for $t$ .

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

Step 6

Simplify the answer.

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Step 6.1

Add $25$ and $5$ .

$30 \cdot 25^{- \frac{1}{2}}$

Step 6.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$30 \cdot \frac{1}{25^{\frac{1}{2}}}$

Step 6.3

Simplify the denominator.

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Step 6.3.1

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

$30 \cdot \frac{1}{{(5^{2})}^{\frac{1}{2}}}$

Step 6.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$30 \cdot \frac{1}{5^{2 (\frac{1}{2})}}$

Step 6.3.3

Cancel the common factor of $2$ .

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Step 6.3.3.1

Cancel the common factor.

$30 \cdot \frac{1}{5^{2 (\frac{1}{2})}}$

Step 6.3.3.2

Rewrite the expression.

$30 \cdot \frac{1}{5^{1}}$

Step 6.3.4

Evaluate the exponent.

$30 \cdot \frac{1}{5}$

Step 6.4

Cancel the common factor of $5$ .

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Step 6.4.1

Factor $5$ out of $30$ .

$5 (6) \cdot \frac{1}{5}$

Step 6.4.2

Cancel the common factor.

$5 \cdot 6 \cdot \frac{1}{5}$

Step 6.4.3

Rewrite the expression.

$6$