The following limit equation is valid:
Prove this in the following two ways:
- Using the definition of the derivative ;
- Using the previous exercise (Section 6.9, Exercise #28).
- Proof. First, we know from Theorem 6.1 part (b) (page 229 of Apostol) that
Therefore, . Then, we recall the definition of the derivative of a function
So, using the definition of the derivative and the fact that for we have ,
Evaluating at we have
since the variable name is unimportant. Thus, we have established the requested formula,
- Proof. From the previous exercise (Section 6.9, Exercise #28) we know
If then as well, so the inequalities still hold with ,
Multiplying all of the terms by (since we may do this without reversing the inequalities),
Since we then have,
Finally, since
we apply the squeeze theorem (Theorem 3.3, page 133 of Apostol) to conclude,
but when x tends to zero from left side?