Calculate the limit for the given function and interval. Verify your answer by using geometry.

Question

Calculate the limit for the given function and interval. Verify your answer by using geometry.

lim𝑁→∞𝐿𝑁 where f(x)=4x+4 over the interval [0,4]

lim𝑁→∞𝐿𝑁=

Answer 1

To calculate the limit of a function as a variable (in this case, N) approaches infinity, you don't need to consider the interval [0, 4] for this specific limit problem. The limit as N approaches infinity is determined solely by the behavior of the function as N becomes extremely large.

For the given function:

f(x) = 4x + 4

The limit as N approaches infinity is:

lim(N → ∞) L(N) = lim(N → ∞) (4N + 4)

To find the limit, you can simply consider the leading term in the function, which is 4N. As N goes to infinity, 4N also goes to infinity, and the constant term 4 does not affect the behavior as N becomes very large. Therefore:

lim(N → ∞) (4N + 4) = ∞

So, the limit as N approaches infinity is infinity.

Now, let's verify this answer geometrically. The graph of the function f(x) = 4x + 4 is a straight line with a slope of 4, passing through the point (0, 4). It extends infinitely in both the positive and negative directions. As x moves towards positive or negative infinity, the value of f(x) becomes larger and larger, approaching infinity.

Geometrically, you can imagine drawing this line and observing how it becomes steeper and approaches infinity as you move away from the origin (x = 0). This behavior confirms the limit result we calculated, which is indeed ∞ as N approaches infinity.