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Chapter 1: Problem 34

Find the limit (if it exists). $$ \lim _{\Delta x \rightarrow 0} \frac{(x+\Delta x)^{2}-x^{2}}{\Delta x} $$

Short Answer

Expert verified
The limit of the given expression as \( \Delta x \rightarrow 0 \) is \( 2x \).

Step by step solution

01

Expand the squared term and simplify the numerator

The first step is to expand the squared term, \( (x + \Delta x)^{2} \). This results in \( x^{2} + 2x\Delta x + \Delta x^{2} \). Substituting this into the expression, we get \(\frac{x^{2} + 2x\Delta x + \Delta x^{2} - x^{2}}{\Delta x}\). Simplifying this gives \(\frac{2x\Delta x + \Delta x^{2}}{\Delta x}\).
02

Simplify the expression by factoring

Next, factor out \( \Delta x \) from the numerator to simplify the equation. This results in the expression becoming \( \Delta x(2x + \Delta x) / \Delta x \).
03

Simplify further by cancelling out terms

Now, we can cancel out the \( \Delta x \) from the numerator and the denominator, giving us the expression \( 2x + \Delta x \).
04

Evaluate the limit

Finally, with the simplified expression, we can evaluate the limit as \( \Delta x \) tends to zero. Taking the limit of \( 2x + \Delta x \) as \( \Delta x \) tends to 0 gives \( 2x + 0 \) or simply \( 2x \).

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