collinito said:Homework Statement
lim[x→-2¯(from the left)] x+2/sqrt(x^2-4)
Homework Equations
The Attempt at a Solution
When I attempted this question I felt that the solution was that this is undefined and when i put it into my ti-89 it said that too. My friend argues that it's actually zero as that's what the leftbound limit approaches graphically. I am curious now to know the answer to this one
collinito said:l'hopitals doesn't work, I already tried it, you still get a zero on the bottom.
The limit for the given function at x=-2 is undefined or does not exist, since the value of the denominator becomes 0 at this point, making the entire expression undefined.
No, the limit cannot be solved by direct substitution since plugging in x=-2 would result in dividing by 0, which is an undefined operation.
The limit can be evaluated by simplifying the expression using algebraic manipulations. In this case, we can factor the denominator to (x+2)(x-2), and cancel out the common factor of (x+2) from the numerator and denominator. This leaves us with the limit of x-2 as x approaches -2 from the left.
Approaching from the left in this limit problem means that we are considering values of x that are slightly smaller than -2, such as -2.0001 or -2.5. This is important because the behavior of the function may be different on the left and right sides of x=-2, and approaching from the left allows us to examine the limit as x approaches -2 from that specific direction.
Yes, L'Hopital's rule can be used to solve this limit problem. By taking the derivative of the numerator and denominator separately, we can rewrite the limit as lim[x→-2¯(from the left)] 1/sqrt(x^2-4). Then, we can plug in x=-2 and evaluate the limit to be -1/2.