If a function is differentiable, the function is continuous. The contrapositive is also true. If a function is not continuous, then it is not differentiable.
A function is differentiable when the limit definition of its derivative exists.
A limit exists when the left and right hand limits are equal.
The limit of sinx/x as x approaches 0 is 1.
The Attempt at a Solution
Okay I first tried making the function continuous but I found that the function would be continuous for 0. I didn't think this was the right answer because the question asked me to use the limit definition so I applied that. Plus, the converse of the statement if a function is diff. it is continuous isn't necessarily true.
The left hand limit is equal to 2. The limit of (2sinx - 0) / (x - 0 ) is 2, using the trig identity mentioned above.
The right hand limit only equals 2 when k = 2. The limit of kx / x as x approaches 0 is k, since the x-terms cancel, and the limit is equal to k. K therefore must equal 2 for the left hand and the right hand limits to be equal.
Is this the correct solution and approach to this proble?
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