I need to figure out what the graph of f(x)= 1 / (2-3x)^3 near x=2/3 would look like.
To determine the graph of f(x) = 1 / (2-3x)^3 near x = 2/3, you can follow these steps:
1. Start by evaluating f(x) at x = 2/3:
f(2/3) = 1 / (2-3(2/3))^3 = 1 / (2-2) = 1 / 0 (undefined)
Since the denominator of the function becomes zero at x = 2/3, the function is undefined at this point. However, we can still explore the behavior of the function as x approaches 2/3 from both sides.
2. Calculate the limit as x approaches 2/3 from the left side:
lim(x→2/3-) f(x) = lim(x→2/3-) 1 / (2-3x)^3
To evaluate this limit, you can use substitution:
lim(x→2/3-) 1 / (2-3x)^3 = 1 / (2-3(2/3))^3 = 1 / (2-2) = 1 / 0 (undefined)
Similarly, the function is undefined at x = 2/3 from the left side.
3. Calculate the limit as x approaches 2/3 from the right side:
lim(x→2/3+) f(x) = lim(x→2/3+) 1 / (2-3x)^3
Again, you can substitute the value of x:
lim(x→2/3+) 1 / (2-3x)^3 = 1 / (2-3(2/3))^3 = 1 / (2-2) = 1 / 0 (undefined)
The function is also undefined at x = 2/3 from the right side.
Based on these calculations, the graph of f(x) = 1 / (2-3x)^3 near x = 2/3 will not have a specific point at x = 2/3. The function will have a vertical asymptote at x = 2/3, indicating that the function values will approach infinity or negative infinity as x gets closer to 2/3 from both sides.