Can anyone help me calculate this?

28. lim
x→∞
(e^x + 1)^(1/)x

2.73

use an intuitive approach

as x --- >∞ , e^x --->∞
so adding 1 to it results in no significant change,
then taking the xth root, will bring you back to e

in effect you are doing (e^x)^(1/x)
= e^(x(1/x)) = e
Kabe approxiated that as 2.73
should have been 2.72 correct to 2 decimals
(2.718281828...)

Sure, I can help you with that! To find the limit of the given expression, we need to apply some properties of limits. Here's how you can calculate it step by step:

Step 1: Start by rewriting the expression as a base raised to a power. In this case, we can rewrite it as:

lim(x→∞) (e^x + 1)^(1/x) = lim(x→∞) [(1 + e^x)^(1/x)]

Step 2: Now, we can rewrite it using the exponential function and the natural logarithm:

lim(x→∞) [(1 + e^x)^(1/x)] = lim(x→∞) [e^(ln(1 + e^x) / x)]

Step 3: Next, we can use the property of limits that states that the limit of a quotient is equal to the quotient of the limits if the limit in the denominator is not zero:

lim(x→∞) [e^(ln(1 + e^x) / x)] = e^(lim(x→∞) [ln(1 + e^x) / x)]

Step 4: Now, let's focus on the expression inside the exponent and calculate its limit separately:

lim(x→∞) [ln(1 + e^x) / x]

Step 5: We can apply L'Hôpital's rule here. Taking the derivative of the numerator and denominator separately:

lim(x→∞) [(1 / (1 + e^x)) * e^x] / 1

This simplifies to:

lim(x→∞) [e^x / (1 + e^x)] = 1

Step 6: Now, let's substitute the result from Step 5 back into the original expression:

e^(lim(x→∞) [ln(1 + e^x) / x)] = e^1 = e

So, the limit of (e^x + 1)^(1/x) as x approaches infinity is e (approximately 2.71828).

Remember to double-check the calculations, as there are several steps involved.