Problem 41 - Algebra and Trigonometry

Question

Find the horizontal asymptote, if there is one, of the graph of each rational function. $$ h(x)=\frac{12 x^{3}}{3 x^{2}+1} $$

Step-by-Step Solution

Answer

Based on the solution above, there are no horizontal asymptotes for this function because as x approaches both positive and negative infinity, the function also approaches positive and negative infinity, respectively.

Step 1: Simplify the function

First, we can simplify the function by factoring out the highest power of x in the numerator and the denominator, which in this case is \(x^{2}\). So, the function becomes: \(h(x) = \frac{{12x(x^{2})}}{{x^{2}(3 + \frac{1}{{x^{2}}})}} = \frac{{12x}}{{3 + \frac{1}{{x^{2}}}}}\).

Step 2: Find limit as x approaches positive infinity

We need to determine the limit of the function as x approaches positive infinity. This is done through observing what happens to the function as x gets larger and larger. For this function, as x approaches infinity, the term \(\frac{1}{{x^{2}}}\) approaches 0 since any number divided by infinity is effectively zero. Therefore the function simplifies to: \(\frac{{12x}}{3}\) or \(4x\). So as x approaches positive infinity, the function approaches \(4 \cdot \infty\) which is also infinity.

Step 3: Find limit as x approaches negative infinity

Similarly, we need to determine the limit of the function as x approaches negative infinity. This is the same process as step 2, but instead we are observing what happens to the function as x gets smaller and smaller. Just like in step 2, as x approaches negative infinity, the term \(\frac{1}{{x^{2}}}\) still approaches 0. So, the function simplifies to \(\frac{{12x}}{3}\) or \(4x\). However, as x approaches negative infinity, the functions approaches \(4 \cdot -\infty\) which is negative infinity.