Compute the limit lim_{x->0} (e^{2x} - 1)/(3x).

Compute the limit lim_{x->0} (e^{2x} - 1)/(3x).

• A. 1/3
• B. 0
• C. 1
• D. 2/3

Answer: (D)

This limit tests the basic exponential limit (e^t − 1)/t → 1 as t → 0 and how constants scale a limit. Write the expression as [(e^{2x} − 1)/(2x)] × (2/3). As x → 0, the first factor approaches 1 because t = 2x → 0. The second factor is just the constant 2/3. Therefore the whole limit is 1 × 2/3 = 2/3. You can also see this quickly with L’Hôpital: differentiate top and bottom to get (2e^{2x})/3, which at x → 0 gives 2/3. So the limit is 2/3.

This limit tests the basic exponential limit (e^t − 1)/t → 1 as t → 0 and how constants scale a limit. Write the expression as [(e^{2x} − 1)/(2x)] × (2/3). As x → 0, the first factor approaches 1 because t = 2x → 0. The second factor is just the constant 2/3. Therefore the whole limit is 1 × 2/3 = 2/3. You can also see this quickly with L’Hôpital: differentiate top and bottom to get (2e^{2x})/3, which at x → 0 gives 2/3. So the limit is 2/3.