Calculus Examples

$lim_{x \to 7} \sqrt{x^{2} + 14 x + 49}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to 7} x^{2} + 14 x + 49}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $7$ .

$\sqrt{lim_{x \to 7} x^{2} + lim_{x \to 7} 14 x + lim_{x \to 7} 49}$

Step 3

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

$\sqrt{{(lim_{x \to 7} x)}^{2} + lim_{x \to 7} 14 x + lim_{x \to 7} 49}$

Step 4

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

$\sqrt{{(lim_{x \to 7} x)}^{2} + 14 lim_{x \to 7} x + lim_{x \to 7} 49}$

Step 5

Evaluate the limit of $49$ which is constant as $x$ approaches $7$ .

$\sqrt{{(lim_{x \to 7} x)}^{2} + 14 lim_{x \to 7} x + 49}$

Step 6

Evaluate the limits by plugging in $7$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

$\sqrt{7^{2} + 14 lim_{x \to 7} x + 49}$

Step 6.2

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

$\sqrt{7^{2} + 14 \cdot 7 + 49}$

Step 7

Simplify the answer.

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Step 7.1

Raise $7$ to the power of $2$ .

$\sqrt{49 + 14 \cdot 7 + 49}$

Step 7.2

Multiply $14$ by $7$ .

$\sqrt{49 + 98 + 49}$

Step 7.3

Add $49$ and $98$ .

$\sqrt{147 + 49}$

Step 7.4

Add $147$ and $49$ .

$\sqrt{196}$

Step 7.5

Rewrite $196$ as $14^{2}$ .

$\sqrt{14^{2}}$

Step 7.6

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

$14$