QuestionFind the limit as approaches 0 for .
Studdy Solution
STEP 1
Assumptions1. We are trying to find the limit as x approaches0 for the function . We are familiar with the limit properties and trigonometric identities3. We know that the limit of a function as x approaches a certain value is the value that the function approaches as x gets closer and closer to that value
STEP 2
First, we notice that as x approaches0, the expression inside the cosine function, , approaches infinity. This is because as x gets closer and closer to0, the denominator of the fraction gets smaller and smaller, making the whole fraction larger and larger.
STEP 3
However, the cosine function oscillates between -1 and1 as its input varies over all real numbers. This means that as the input to the cosine function gets larger and larger (or smaller and smaller), the output of the function continues to oscillate between -1 and1.
STEP 4
Therefore, as x approaches0, the expression does not approach a single value, but instead oscillates between -1 and1.
STEP 5
In mathematical terms, we say that the limit of the function as x approaches0 does not exist. This is because the function does not approach a single value as x approaches0, but instead continues to oscillate between -1 and1.
So, the solution to the problem is that the limit does not exist.
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