Limits, Continuity And Differentiability

Given the function: and that: for all .We need to determine the value of: First, compute the derivative of : Thus: To find , match with: Introduce the substitution , so: This shows that: The derivative at 0 is thus: Therefore, the value of is 1.

If a curve is smooth at a point, it has no sharp turns or discontinuities, meaning a consistent tangent can be drawn. This ensures continuity and a unique slope at the point, fulfilling the conditions for differentiability. Thus, smoothness at a point guarantees the curve is differentiable there.Shortcut/advice for imagining graph: Here, The makes the graph mirror itself about the -axis(. As is the same around -axis, it behaves the same as .

When considering whether a function is continuous based on the continuity of another function , where is expressed in terms of , we evaluate several scenarios:Case: For to be continuous, must also be continuous. This is because the composition of continuous functions (in this case, the cubic function and ) results in a continuous function.Case: Consider the function: Here, is continuous and equals 2 for any value of , even though itself is not continuous due to the jump discontinuity at .Case: Using the same function as in the previous case: Here, is continuous and equals 4 for any value of , while is not continuous.Case: Consider the function: In this case, is continuous and equals 1 for any value of , whereas is not continuous due to the jump at .In summary, while is continuous in the forms evaluated, only in the case of does this directly imply itself is continuous. The other forms allow to be continuous even if is not.