Sorry I have another question, but it's a true/false question and it's asking
"if Lim x->5 (m(x)-m(5))/(x-5) = -3, then m(x) is continuous at x=5."
Can't there be a hole, so would it be false?
correct. m(5) must be defined and equal to the limit.
looks like [ m(5+e) - m(5) ] / ( x-5) = -3 as e--->0
looks like the derivative dm/dx = -3 at x = 5
If we can define a unique derivative(slope) there, m better be continuous there.
Sketch graphs of these problems :)
To determine if the statement is true or false, we need to consider the definition and properties of continuity.
For a function to be continuous at a point, it needs to satisfy three conditions:
1. The function must be defined at that point.
2. The limit of the function as x approaches that point must exist.
3. The value of the function at that point must be equal to the limit.
In this case, we are given the expression:
Lim x->5 (m(x)-m(5))/(x-5) = -3
From this expression, we can observe that the limit of the left-hand side exists and is equal to -3. This implies that condition 2 is satisfied, i.e., the limit of m(x) as x approaches 5 exists.
However, to determine whether m(x) is continuous at x=5, we need to consider condition 1 and condition 3 as well.
Condition 1:
The function must be defined at x=5. In this case, we are not given any specific information about the function m(x) at x=5. Therefore, we cannot determine if condition 1 is satisfied.
Condition 3:
The value of the function at x=5 must be equal to the limit. Again, we are not given any specific information about the value of m(x) at x=5. So we cannot determine if condition 3 is satisfied.
Based on the information given, we cannot conclude definitively whether m(x) is continuous at x=5. Therefore, the statement "if Lim x->5 (m(x)-m(5))/(x-5) = -3, then m(x) is continuous at x=5" is indeterminate.
To make a definitive conclusion, we would need more information about the function m(x) and its behavior at x=5.