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# In two different ways, prove the limit as x tends to 0 of (log (1+x))/x is 1

The following limit equation is valid:

Prove this in the following two ways:

1. Using the definition of the derivative ;
2. Using the previous exercise (Section 6.9, Exercise #28).

1. 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,

2. 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,