Given that

 lim *x*→2 *f*(*x*) = 4    lim *x*→2 *g*(*x*) = −2    lim *x*→2 *h*(*x*) = 0,

find the limits, if they exist. (If an answer does not exist, enter DNE.) (e)    

lim *x*→2 

*g*(*x*)*/h*(*x*)


We are tasked with evaluating:

\[
\lim_{x \to 2} \frac{g(x)}{h(x)}
\]

Given that:
- \(\lim_{x \to 2} g(x) = -2\)
- \(\lim_{x \to 2} h(x) = 0\)

Now, we substitute these limit values into the expression:

\[
\lim_{x \to 2} \frac{g(x)}{h(x)} = \frac{-2}{0}
\]

Since we are dividing a non-zero number by zero, this results in an undefined limit. Specifically, division by zero typically leads to an infinite limit, but whether it is positive or negative infinity depends on the behavior of \(h(x)\) as \(x\) approaches 2 from both sides. However, since we're not given any information about whether \(h(x)\) approaches zero from the positive or negative direction, we cannot conclude whether the result is \(+\infty\) or \(-\infty\).

Thus, the answer is **DNE** (Does Not Exist).

Would you like a detailed breakdown of why division by zero leads to this result?

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Here are 5 related questions for further exploration:
1. What happens when you divide by zero in a limit if the numerator is also zero?
2. How can we apply L'Hopital's Rule to indeterminate forms like \(\frac{0}{0}\)?
3. What conditions guarantee that a limit involving division does not exist?
4. Can you think of a case where dividing by zero gives a finite value in limits?
5. What is the significance of one-sided limits in understanding division by zero?

**Tip**: When dividing by a function that tends to zero, always check the sign of the denominator as you approach the limit to avoid undefined results!

Given that lim x→2 f(x) = 4, lim x→2 g(x) = −2, lim x→2 h(x) = 0, find the limits, if they exist. (If an answer does not exist, enter DNE.) lim x→2 g(x)/h(x).

Solve the limit problem lim x→2 g(x)/h(x) where g(x) = -2 and h(x) = 0. Explore key concepts like indeterminate forms and division by zero in limits. Learn how the undefined result leads to DNE and understand the importance of checking the behavior of functions near singular points. Suitable for Grades 10-12.

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