For a function to be continuous at a point, it means that the limit of the function as x approaches that point is equal to the value of the function at that point. In other words, there are no abrupt jumps or breaks in the graph of the function at that point.
The limit of 2x^2+5 as x approaches 3 is 23. This can be calculated by substituting the value of 3 for x in the function and solving for the resulting expression, which is 23 in this case.
To prove that 2x^2+5 is continuous at x=3, you can use the definition of continuity and show that the limit of the function as x approaches 3 is equal to the value of the function at x=3. This can be done by substituting 3 for x in the function and showing that the resulting expression is equal to 23.
There are three types of discontinuities - removable, jump, and infinite. A removable discontinuity occurs when there is a hole in the graph of the function at a particular point. A jump discontinuity occurs when there is a sudden jump or gap in the graph at a point. An infinite discontinuity occurs when the limit of the function at a point is either positive or negative infinity.
No, not every continuous function is differentiable. A function is differentiable at a point if its derivative exists at that point. There are certain types of functions, such as those with corners or sharp turns, that are continuous but not differentiable at certain points. However, if a function is differentiable at a point, it must also be continuous at that point.