Evaluate the Limit

Problem

(lim_x→0)((tan(x)−x)/(x3))

Solution

1. Identify the indeterminate form by substituting x=0 into the expression, which yields 0/0

2. Apply L'Hôpital's Rule by differentiating the numerator and the denominator with respect to x

(lim_x→0)((d(tan(x))−x)/d(x)/d(x3)/d(x))

(lim_x→0)((sec2(x)−1)/(3*x2))

3. Simplify the expression using the trigonometric identity sec2(x)−1=tan2(x)

(lim_x→0)(tan2(x)/(3*x2))

4. Rewrite the limit to isolate the known fundamental limit (lim_x→0)(tan(x)/x)=1

1/3*(lim_x→0)(tan(x)/x)

5. Evaluate the limit by substituting the known value.

1/3*(1)2=1/3

Final Answer

(lim_x→0)((tan(x)−x)/(x3))=1/3

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