Can someone please help explain these two questions for me. For the first one my guess is that it is -infinity and for the second question it would be at 0.
Find the limit.
lim_(x->infinity) (x^(3) - 2x + 3)/(4 - 5 x^(2))
Locate the discontinuities of the function.
y = 2/(1 + e^(1\/x))
Sure! Let me explain how to find the answers to both questions.
1. Find the limit:
To find the limit of a function as x approaches infinity, we can analyze the behavior of the function when x becomes very large.
In this case, the function is (x^3 - 2x + 3)/(4 - 5x^2). To find the limit as x approaches infinity, we need to consider the highest power of x in the numerator and denominator.
The highest power of x in the numerator is x^3, and the highest power of x in the denominator is x^2. Since the degree of the numerator is higher than the degree of the denominator, we can conclude that as x approaches infinity, the function will also approach infinity.
Therefore, the limit as x approaches infinity of the given function is positive infinity.
2. Locate the discontinuities of the function:
To find the discontinuities of a function, we need to determine the values of x that make the function undefined.
In the given function, y = 2/(1 + e^(1/x)), there's a potential discontinuity when the denominator becomes zero.
To find the values of x that make the denominator zero, we set 1 + e^(1/x) = 0 and solve for x. However, in this case, it is not possible for the denominator to be zero because e^(1/x) is always positive for any real value of x.
Therefore, there are no discontinuities in the function y = 2/(1 + e^(1/x)). It is continuous for all real values of x.
I hope this explanation helps! Let me know if you have any further questions.