Evaluate the Limit ( limit as x approaches 0 of sin(x^2))/x

Problem

(lim_x→0)(sin(x2)/x)

Solution

1. Identify the form of the limit by substituting x=0 into the expression.

2. Evaluate the numerator and denominator at x=0 which results in sin(0)/0=0/0 an indeterminate form.

3. Apply L'Hôpital's Rule, which states that (lim_x→c)(ƒ(x)/g(x))=(lim_x→c)((ƒ(x)′)/(g(x)′)) if the limit is indeterminate.

4. Differentiate the numerator sin(x2) using the chain rule to get 2*x*cos(x2)

5. Differentiate the denominator x to get 1

6. Rewrite the limit using these derivatives.

(lim_x→0)((2*x*cos(x2))/1)

7. Substitute x=0 into the new expression to find the limit.

2*(0)*cos(0)=0⋅1=0

Final Answer

(lim_x→0)(sin(x2)/x)=0

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