Calculus Examples

$\frac{lim_{t \to 4} t - \sqrt{3 t + 4}}{4 - t}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $4$ .

$\frac{lim_{t \to 4} t - lim_{t \to 4} \sqrt{3 t + 4}}{4 - t}$

Step 2

Move the limit under the radical sign.

$\frac{lim_{t \to 4} t - \sqrt{lim_{t \to 4} 3 t + 4}}{4 - t}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $4$ .

$\frac{lim_{t \to 4} t - \sqrt{lim_{t \to 4} 3 t + lim_{t \to 4} 4}}{4 - t}$

Step 4

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

$\frac{lim_{t \to 4} t - \sqrt{3 lim_{t \to 4} t + lim_{t \to 4} 4}}{4 - t}$

Step 5

Evaluate the limit of $4$ which is constant as $t$ approaches $4$ .

$\frac{lim_{t \to 4} t - \sqrt{3 lim_{t \to 4} t + 4}}{4 - t}$

Step 6

Evaluate the limits by plugging in $4$ for all occurrences of $t$ .

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Step 6.1

Evaluate the limit of $t$ by plugging in $4$ for $t$ .

$\frac{4 - \sqrt{3 lim_{t \to 4} t + 4}}{4 - t}$

Step 6.2

Evaluate the limit of $t$ by plugging in $4$ for $t$ .

$\frac{4 - \sqrt{3 \cdot 4 + 4}}{4 - t}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Multiply $3$ by $4$ .

$\frac{4 - \sqrt{12 + 4}}{4 - t}$

Step 7.1.2

Add $12$ and $4$ .

$\frac{4 - \sqrt{16}}{4 - t}$

Step 7.1.3

Rewrite $16$ as $4^{2}$ .

$\frac{4 - \sqrt{4^{2}}}{4 - t}$

Step 7.1.4

Pull terms out from under the radical, assuming positive real numbers.

$\frac{4 - 1 \cdot 4}{4 - t}$

Step 7.1.5

Multiply $- 1$ by $4$ .

$\frac{4 - 4}{4 - t}$

Step 7.1.6

Subtract $4$ from $4$ .

$\frac{0}{4 - t}$

Step 7.2

Divide $0$ by $4 - t$ .

$0$