A limit represents the y-value of a graph as x approaches a given number from both sides. A limit does not exist if the y-value is different depending on whether x is approached from the left or right. If a function is continuous, the limit value equals the function value. L'Hopital's Rule is used when applying the substitution method would result in an undefined form. A function cannot be differentiable on all real numbers if it is not continuous, but it can be differentiable over continuous intervals.