Answer :

(a) The limit of (x + tan(x)) / sin(x) as x approaches 0 is 1.

(b) The limit of (3x / (x - 1)) - (3 / ln(x)) as x approaches infinity is 3.

(a) To find the limit of (x + tan(x)) / sin(x) as x approaches 0, we can apply l'Hôpital's Rule. Both the numerator and the denominator approach 0 as x approaches 0. Taking the derivative of the numerator and denominator with respect to x, we get:

Numerator derivative: 1 + sec^2(x)

Denominator derivative: cos(x)

Now, substituting x = 0, we get:

Numerator derivative at x = 0: 1

Denominator derivative at x = 0: 1

So, the limit becomes 1 / 1, which is equal to 1.

(b) For the limit of (3x / (x - 1)) - (3 / ln(x)) as x approaches infinity, we again use l'Hôpital's Rule. The denominator (x - 1) and the natural logarithm ln(x) approach infinity as x approaches infinity. Taking the derivatives:

Derivative of numerator: 3

Derivative of denominator (x - 1): 1

Derivative of denominator ln(x): 1/x

At infinity, both the derivatives become constants (3 and 1), and the limit becomes:

(3) / (1) - (3) / (infinity) = 3 - 0 = 3.

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