WHAT IS THE LIMIT WHEN X TOWARDS TO 0 OF sin6x/5x ? lim𝑥→0 𝑠𝑖𝑛6𝑥/5𝑥

To find the limit of the expression \(\frac{\sin(6x)}{5x}\) as \(x\) approaches 0, we can use a fundamental limit property of sine. Specifically, we know that as \(x\) approaches 0, \(\frac{\sin(x)}{x}\) approaches 1. We can leverage this property to solve our limit problem.

Breaking Down the Limit

We start with the limit:

\(\lim_{x \to 0} \frac{\sin(6x)}{5x}\)

To apply the known limit property, we need to manipulate the expression so that we can use \(\frac{\sin(x)}{x}\). We can rewrite our limit as follows:

\(\lim_{x \to 0} \frac{\sin(6x)}{6x} \cdot \frac{6}{5}\)

Step-by-Step Explanation

• First, notice that we can factor out the constant \(6\) from the sine function. This gives us \(\frac{\sin(6x)}{6x}\), which is a form we can work with.
• Next, we know from the limit property that \(\lim_{u \to 0} \frac{\sin(u)}{u} = 1\). In our case, as \(x\) approaches 0, \(6x\) also approaches 0. Thus, we can apply this limit property here.
• Now, we can express our limit as: \(\lim_{x \to 0} \frac{\sin(6x)}{6x} \cdot \frac{6}{5} = 1 \cdot \frac{6}{5}\).

Final Calculation

Putting it all together, we find:

\(\lim_{x \to 0} \frac{\sin(6x)}{5x} = \frac{6}{5}\)

Thus, the limit of \(\frac{\sin(6x)}{5x}\) as \(x\) approaches 0 is \(\frac{6}{5}\). This result shows how we can use known limits to simplify and solve more complex limit problems effectively.