Evaluate the Limit limit as h approaches 0 of (2xh+h^2)/h

Problem

(lim_h→0)((2*x*h+h2)/h)

Solution

1. Factor the numerator by taking out the greatest common factor, which is h

(2*x*h+h2)/h=(h*(2*x+h))/h

2. Simplify the expression by canceling the common factor h from the numerator and the denominator, noting that h≠0 as it approaches the limit.

(h*(2*x+h))/h=2*x+h

3. Evaluate the limit by substituting h=0 into the simplified expression.

(lim_h→0)(2*x+h)=2*x+0

4. Simplify the result to find the final value.

2*x+0=2*x

Final Answer

(lim_h→0)((2*x*h+h2)/h)=2*x

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