Evaluate the limit as x approaches 0 of (e^(x2) - cos(x)) / (x2).
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Step-by-Step Solution
Step 1: Check for indeterminate form
First, we substitute x=0 into the numerator and the denominator to check the form of the limit. We find that both the numerator and the denominator approach 0 resulting in an indeterminate form of 00. This indicates that L'Hôpital's Rule can be applied.
Step 2: Apply L'Hôpital's Rule
Since the limit is in the indeterminate form 00 we can apply L'H o^ pital's Rule. This rule states that if limx→cg(x)f(x) is of the form 00or∞∞, then limx→cg(x)f(x)=limx→cg′(x)f′(x), provided the latter limit exists.
Step 3: Differentiate numerator and denominator
We differentiate the numerator and the denominator. The derivative of ex2 is ex2⋅(2x) by the chain rule, and the derivative of - cos(x) is -(- sin(x)) = sin(x) The derivative of x 2is2x.
Step 4: Substitute derivatives and re-evaluate limit
Now we substitute the derivatives back into the limit expression. We then try to evaluate the limit again by substituting x=0 We find that the numerator becomes 2(0)e02+sin(0)=0+0=0, and the denominator becomes 2(0) = 0 This is still an indeterminate form 00, so we must apply L'Hôpital's Rule again.
Step 5: Apply L'Hôpital's Rule again
We differentiate the new numerator and denominator. For 2xex2 we use the product rule: 2ex2+2x(ex2⋅2x)=2ex2+4x2ex2. The derivative of sin(x) is cos(x). The derivative of 2x is 2.
Step 6: Evaluate the final limit
Finally, we substitute x=0 into the expression with the second derivatives. The numerator becomes 2e0+4(0)2e0+cos(0)=2(1)+0+1=3. The denominator is 2. Thus, the limit is 23.