Calculus Examples

$\frac{lim_{x \to 8} \sqrt{x + e^{4 x}}}{e^{2 x} + x}$

Step 1

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8} x + e^{4 x}}}{e^{2 x} + x}$

Step 2

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

$\frac{\sqrt{lim_{x \to 8} x + lim_{x \to 8} e^{4 x}}}{e^{2 x} + x}$

Step 3

Move the limit into the exponent.

$\frac{\sqrt{lim_{x \to 8} x + e^{lim_{x \to 8} 4 x}}}{e^{2 x} + x}$

Step 4

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{lim_{x \to 8} x + e^{4 lim_{x \to 8} x}}}{e^{2 x} + x}$

Step 5

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

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Step 5.1

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

$\frac{\sqrt{8 + e^{4 lim_{x \to 8} x}}}{e^{2 x} + x}$

Step 5.2

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

$\frac{\sqrt{8 + e^{4 \cdot 8}}}{e^{2 x} + x}$

Step 6

Multiply $4$ by $8$ .

$\frac{\sqrt{8 + e^{32}}}{e^{2 x} + x}$