What is the limit as x approaches 0 of sqrt(e^{2x)+e^x}-sqrt(e^{2x)+x} ?

The limit as x approaches 0 of sqrt(e^{2x)+e^x}-sqrt(e^{2x)+x} is sqrt(2)-1

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limit as x approaches 0 of sqrt(e^{2x)+e^x}-sqrt(e^{2x)+x}

x \to 0 lim (e^{2 x} + e^{x} - e^{2 x} + x)

2 - 1

Decimal

0.41421 \dots

Solution steps

x \to 0 lim (e^{2 x} + e^{x} - e^{2 x} + x)

Plug in the value

x = 0

= e^{2 \cdot 0} + e^{0} - e^{2 \cdot 0} + 0

Simplify

e^{2 \cdot 0} + e^{0} - e^{2 \cdot 0} + 0 : 2 - 1

= 2 - 1

What is the limit as x approaches 0 of sqrt(e^{2x)+e^x}-sqrt(e^{2x)+x} ?
The limit as x approaches 0 of sqrt(e^{2x)+e^x}-sqrt(e^{2x)+x} is sqrt(2)-1