Question

Найти предел (3^(x)-81)/(sin(pi*x)) при x -> 4

Answer 1

Решаем по правилу Лопиталя

Answer 2

To find the limit of the expression as x approaches 4, we can directly substitute x = 4 into the expression and simplify. However, we need to make sure that the expression is well-defined at x = 4, as dividing by sin(pix) can cause issues. We'll check if sin(pix) is equal to zero at x = 4.

Since sin(pix) has a period of 1 (sin(pix) = sin(pi*(x + k)), where k is any integer), we can find the limit as x approaches 4 from both sides (x → 4⁻ and x → 4⁺) to check if sin(pi*x) is zero at x = 4.

1. Limit as x approaches 4 from the left (x → 4⁻):

lim(x → 4⁻) [ (3^x - 81) / sin(pi*x) ] = (3^4 - 81) / sin(pi * 4)

2. Limit as x approaches 4 from the right (x → 4⁺):

lim(x → 4⁺) [ (3^x - 81) / sin(pi*x) ] = (3^4 - 81) / sin(pi * 4)

Since both expressions have the same value, we can evaluate either one of them. Let's use the left-side limit for simplicity:

lim(x → 4⁻) [ (3^x - 81) / sin(pi*x) ] = (3^4 - 81) / sin(pi * 4)

Now, let's calculate the value:

3^4 = 81 sin(pi * 4) = sin(4 * pi) = 0 (since sin(2 * n * pi) = 0, where n is an integer)

The denominator is zero, which means the expression is undefined at x = 4.

In conclusion, the limit of (3^x - 81) / sin(pix) as x approaches 4 does not exist because the denominator sin(pix) becomes zero at x = 4.