Dick said:What does continuous mean in terms of limits? k(x)=(-1) means the function is constant. k(4)=2 contradicts the previous statement. What's the real problem?
chukie said:sorry i typed the question wrong k(3)=-1
Dick said:S'ok. But the question is still odd, because the values of the function don't have anything to do with whether the two limits are equal. If a function is continuous at x=3, what can you say about it's left and right hand limits? Do you mean to say k(x) is only defined on the interval [3,4]? Or is it defined and continuous everywhere?
The limit of a function k(x) at x=3 refers to the value that the function approaches as the input value x gets closer and closer to 3. It does not necessarily mean that the function will have a value at x=3, but rather the behavior of the function as it approaches that point.
The limit of a function at x=3 is typically calculated by evaluating the function at values very close to 3, both from the left and right sides. If the values approach the same number, then that is the limit. However, if the values approach different numbers, then the limit does not exist.
If the limit of a function at x=3 does not exist, it means that the function does not have a defined value at x=3. This could be due to a jump or discontinuity in the function, or the function approaching different values from the left and right sides of x=3.
Limits are important in mathematics because they allow us to understand the behavior of functions at specific points. They also help us solve complex problems by breaking them down into smaller parts and analyzing the behavior of the function at each point. Limits are also essential in calculus, as they are used to define derivatives and integrals.
Yes, the limit of a function at x=3 can be different from the actual value of the function at x=3. This is because the limit only considers the behavior of the function as it approaches x=3, not the actual value at that point. The function may have a discontinuity or be undefined at x=3, but still have a well-defined limit at that point.