Calculus Examples

$lim_{z \to 2} \sqrt{7 z^{2} + 14 z - 7}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{z \to 2} 7 z^{2} + 14 z - 7}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $z$ approaches $2$ .

$\sqrt{lim_{z \to 2} 7 z^{2} + lim_{z \to 2} 14 z - lim_{z \to 2} 7}$

Step 3

Move the term $7$ outside of the limit because it is constant with respect to $z$ .

$\sqrt{7 lim_{z \to 2} z^{2} + lim_{z \to 2} 14 z - lim_{z \to 2} 7}$

Step 4

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

$\sqrt{7 {(lim_{z \to 2} z)}^{2} + lim_{z \to 2} 14 z - lim_{z \to 2} 7}$

Step 5

Move the term $14$ outside of the limit because it is constant with respect to $z$ .

$\sqrt{7 {(lim_{z \to 2} z)}^{2} + 14 lim_{z \to 2} z - lim_{z \to 2} 7}$

Step 6

Evaluate the limit of $7$ which is constant as $z$ approaches $2$ .

$\sqrt{7 {(lim_{z \to 2} z)}^{2} + 14 lim_{z \to 2} z - 1 \cdot 7}$

Step 7

Evaluate the limits by plugging in $2$ for all occurrences of $z$ .

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Step 7.1

Evaluate the limit of $z$ by plugging in $2$ for $z$ .

$\sqrt{7 \cdot 2^{2} + 14 lim_{z \to 2} z - 1 \cdot 7}$

Step 7.2

Evaluate the limit of $z$ by plugging in $2$ for $z$ .

$\sqrt{7 \cdot 2^{2} + 14 \cdot 2 - 1 \cdot 7}$

Step 8

Simplify the answer.

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Step 8.1

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

$\sqrt{7 \cdot 4 + 14 \cdot 2 - 1 \cdot 7}$

Step 8.2

Multiply $7$ by $4$ .

$\sqrt{28 + 14 \cdot 2 - 1 \cdot 7}$

Step 8.3

Multiply $14$ by $2$ .

$\sqrt{28 + 28 - 1 \cdot 7}$

Step 8.4

Multiply $- 1$ by $7$ .

$\sqrt{28 + 28 - 7}$

Step 8.5

Add $28$ and $28$ .

$\sqrt{56 - 7}$

Step 8.6

Subtract $7$ from $56$ .

$\sqrt{49}$

Step 8.7

Rewrite $49$ as $7^{2}$ .

$\sqrt{7^{2}}$

Step 8.8

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

$7$