Calculus Examples

$lim_{x \to 8} (\sqrt{(x + m) (x + n)}) - x$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$lim_{x \to 8} \sqrt{(x + m) (x + n)} - lim_{x \to 8} x$

Step 2

Move the limit under the radical sign.

$\sqrt{lim_{x \to 8} (x + m) (x + n)} - lim_{x \to 8} x$

Step 3

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $8$ .

$\sqrt{lim_{x \to 8} x + m \cdot lim_{x \to 8} x + n} - lim_{x \to 8} x$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\sqrt{(lim_{x \to 8} x + lim_{x \to 8} m) \cdot lim_{x \to 8} x + n} - lim_{x \to 8} x$

Step 5

Evaluate the limit of $m$ which is constant as $x$ approaches $8$ .

$\sqrt{(lim_{x \to 8} x + m) \cdot lim_{x \to 8} x + n} - lim_{x \to 8} x$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\sqrt{(lim_{x \to 8} x + m) \cdot (lim_{x \to 8} x + lim_{x \to 8} n)} - lim_{x \to 8} x$

Step 7

Evaluate the limit of $n$ which is constant as $x$ approaches $8$ .

$\sqrt{(lim_{x \to 8} x + m) \cdot (lim_{x \to 8} x + n)} - lim_{x \to 8} x$

Step 8

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\sqrt{(8 + m) \cdot (lim_{x \to 8} x + n)} - lim_{x \to 8} x$

Step 8.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\sqrt{(8 + m) \cdot (8 + n)} - lim_{x \to 8} x$

Step 8.3

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\sqrt{(8 + m) \cdot (8 + n)} - 8$