Question

lim e^(1+h)-e/h

h->0

Answer 1

[AI-generated answer — Explain Bot]

To find the limit of the expression as h approaches 0, plug in 0 for h and simplify the expression.

lim(h->0) (e^(1+h) - e)/h

Step 1: Substitute 0 for h:

(e^(1+0) - e)/0

Step 2: Simplify the expression:

(e^1 - e)/0

Step 3: Since we have a denominator of 0, we cannot directly evaluate the limit.

However, we can rewrite the expression to get a better understanding of the limit. Notice that the expression (e^1 - e) is equal to 0, so we have:

0/0

This is an indeterminate form. Indeterminate forms occur when we get conflicting results or encounter an undefined operation.

To further evaluate the limit in this case, we can use L'Hôpital's Rule, which states that if we have a limit of the form 0/0 or ∞/∞, and the limit exists, then we can take the derivative of the numerator and denominator separately and evaluate the limit again.

Step 4: Apply L'Hôpital's Rule by taking the derivative of the numerator and denominator:

lim(h->0) (e^(1+h) - e)/h = lim(h->0) (d/dh(e^(1+h) - e))/(d/dh(h))

Step 5: Evaluate the derivatives:

lim(h->0) (e^(1+h) - e)/(1)

Since the derivative of e^(1+h) is e^(1+h), the derivative of e^(1+h) - e is e^(1+h).

Step 6: Substitute h = 0 to find the limit:

lim(h->0) (e^(1+0) - e)/(1) = (e^1 - e)/(1) = (e - e)/1 = 0/1 = 0

Therefore, the limit of the expression as h approaches 0 is 0.