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Problem 10

# Determine if the given limit leads to a determinate or indeterminate form. Evaluate the limit if it exists, or say why if not. $$\lim _{x \rightarrow 0} \frac{2 x^{2}+4}{x}$$

Expert verified
The given limit, $$\lim _{x \rightarrow 0} \frac{2 x^{2}+4}{x}$$, leads to an indeterminate form. Since there are no common terms in the numerator and denominator to factor out and simplify, the limit does not exist due to the division by zero.
See the step by step solution

## Step 1: Determine the form of the limit

First, we need to see if the limit leads to a determinate or indeterminate form. To do this, we plug in x = 0 into the expression: $\frac{2(0)^2 + 4}{0} = \frac{4}{0}$ Since division by zero is undefined, the limit is indeterminate.

## Step 2: Simplify the expression

Next, let's try to simplify the expression in the limit to see if we can evaluate the limit: $\frac{2x^2+4}{x} = \frac{2(x^2+2)}{x}$ Now, we want to factor out the common terms in the numerator and denominator. As there are no common terms to factor out in this situation, we can't simplify the expression further.

## Step 3: Conclusion

Due to the indeterminate form of the limit and the inability to simplify the expression further, the limit does not exist. This is because of the division by zero.

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