Locate the discontinuities of the function
y=(5)/(3+e^(1/x))
first of all you dont need () those for the five so that's one, and you dont need the () for the 1/x
your problem should look like this
y=5(3+e^1x)
either that that idk
you do need the () around 1/x, since powers are done first, and you don't want (e^1)/x
3+e^(1/x) is never zero, so there are no discontinuities because of a zero denominator.
However, since 1/x is undefined at x=0, there is a jump there.
lim as x->0- = 5/3
lim as x->0+ = 0
To find the discontinuities of the function y = (5)/(3 + e^(1/x)), we need to identify values of x where the function is not defined or where it exhibits certain characteristics that make it discontinuous.
Step 1: Determine the conditions for the function to be undefined.
In this case, the function is undefined when the denominator (3 + e^(1/x)) equals zero because division by zero is undefined. Therefore, we need to find the values of x that make the denominator zero.
3 + e^(1/x) = 0
Step 2: Solve the equation to find the values of x that make the denominator zero.
To solve this equation, we need to isolate the term containing the variable.
e^(1/x) = -3
Step 3: Take the natural logarithm (ln) of both sides of the equation to eliminate the exponential term.
ln(e^(1/x)) = ln(-3)
1/x = ln(-3)
Step 4: Solve for x.
To solve for x, we need to isolate the variable.
x = 1/ln(-3)
Step 5: Analyze the result and determine if there are any discontinuities.
Since we have obtained a numeric value for x (1/ln(-3)), there are no discontinuities due to this equation.
However, it is important to note that the function y = (5)/(3 + e^(1/x)) might exhibit some behaviors that could lead to discontinuities. For instance, the exponential term e^(1/x) becomes infinitely large (approaches infinity) as x approaches 0 from the left side (x → 0-). As a result, the denominator (3 + e^(1/x)) approaches infinity, and the function y approaches 0. This behavior does not constitute a discontinuity, but it indicates a limit.
In summary, there are no explicit values of x that make the function undefined. However, the function exhibits behavior close to x = 0, leading to a limit without a distinct value at that point.