Evaluate the Limit

Problem

(lim_h→0)(((x+h)2−x2)/h)

Solution

1. Expand the squared binomial (x+h)2 in the numerator.

(x+h)2=x2+2*x*h+h2

2. Substitute the expanded form back into the limit expression.

(lim_h→0)((x2+2*x*h+h2−x2)/h)

3. Simplify the numerator by canceling the x2 and −x2 terms.

(lim_h→0)((2*x*h+h2)/h)

4. Factor out the common factor of h from the numerator.

(lim_h→0)((h*(2*x+h))/h)

5. Divide the numerator and denominator by h to remove the indeterminate form.

(lim_h→0)(2*x+h)

6. Evaluate the limit by substituting h=0 into the remaining expression.

2*x+0=2*x

Final Answer

(lim_h→0)(((x+h)2−x2)/h)=2*x

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