Question

Limits Evaluate the following limits. Use l'Hópital's Rule when it is comvenient and applicable. $$\lim _{x \rightarrow 0} \csc 6 x \sin 7 x$$

Short answer

Answer: The limit of the function \( \csc 6x \sin 7x\) as \(x\) approaches \(0\) is \(\frac{7}{6}\).

Step-by-step solution

Rewrite the function in the form f(x)/g(x)

First, recall that \(\csc x = \frac{1}{\sin x}\). Rewrite the given expression, using this definition: $$ \lim_{x \rightarrow 0} \csc 6x \sin 7x = \lim_{x \rightarrow 0} \frac{1}{\sin 6x} \cdot \sin 7x = \lim_{x \rightarrow 0} \frac{\sin 7x}{\sin 6x}$$ Now, the function has the form \(\frac{f(x)}{g(x)}\), with \(f(x) = \sin 7x\) and \(g(x) = \sin 6x\).

Check if both f(x) and g(x) approach 0 or infinity at the given limit

To apply L'Hôpital's Rule, both \(f(x)\) and \(g(x)\) must approach \(0\) or \(\pm \infty\) at the given limit \(x \rightarrow 0\). Check the functions: $$\lim_{x \rightarrow 0} \sin 7x = \sin (7 \cdot 0) = 0$$ $$\lim_{x \rightarrow 0} \sin 6x = \sin (6 \cdot 0) = 0$$ Both functions satisfy this condition, so we can use L'Hôpital's Rule.

Apply L'Hôpital's Rule: Compute the derivatives of f(x) and g(x)

To apply L'Hôpital's Rule, compute the derivatives of \(f(x)\) and \(g(x)\): $$f'(x) = \frac{d}{dx} \sin 7x = 7 \cos 7x$$ $$g'(x) = \frac{d}{dx} \sin 6x = 6 \cos 6x$$

Apply L'Hôpital's Rule: Compute the limit of the derivatives

Now calculate the new limit with the derivatives of \(f(x)\) and \(g(x)\): $$\lim_{x \rightarrow 0} \frac{f'(x)}{g'(x)} = \lim_{x \rightarrow 0} \frac{7 \cos 7x}{6 \cos 6x}$$ The limit of the numerator is \(7 \cos (7 \cdot 0) = 7\) and the limit of the denominator is \(6 \cos (6 \cdot 0) = 6\).

Final result

Now, substitute these values to get the limit: $$\lim_{x \rightarrow 0} \frac{7 \cos 7x}{6 \cos 6x} = \frac{7}{6}$$ Hence, the limit of the given function is: $$\lim_{x \rightarrow 0} \csc 6x \sin 7x = \frac{7}{6}$$

Key concepts

Limits in Calculus

When studying calculus, one of the fundamental concepts you'll encounter is the limit. Limits help us understand the behavior of functions as they approach a specific point. It's like trying to find out what value a function is getting closer to, without actually reaching that point.

For example, the expression \(\lim_{x \rightarrow a} f(x)\) essentially asks: As 'x' gets closer and closer to 'a', what value does 'f(x)' approach? In some cases, 'x' may approach a number that makes 'f(x)' undefined, like dividing by zero.

But with the magical rule of L'Hôpital, also spelled as L'Hospital's Rule, we can often resolve these indeterminate forms, such as \(0/0\) or \(\infty/\infty\), by taking the derivative of the numerator and the denominator separately and then evaluating the limit again. Just remember, to use L'Hôpital's Rule, both the numerator and the denominator must approach zero or infinity as 'x' approaches the point in question.

Sine Function Properties

The sine function, written as \(\sin(x)\), is one of the basic trigonometric functions. It has several intriguing properties that are quite useful in calculus:

• The sine function is periodic, meaning it repeats its values in regular intervals. For \(\sin(x)\), this interval is \(2\pi\), which is the sine wave's period.
• Sine is an odd function, so \(\sin(-x) = -\sin(x)\). When graphed, it is symmetric about the origin.
• The range of the sine function is between -1 and 1, inclusive. This means that no matter what value of 'x' you plug in, the output will always lie within this range.
• At \(x = 0\), the sine function equals 0, which is an essential aspect when evaluating limits involving sine at zero.

Understanding these properties allows students to predict the behavior of the sine function under various transformations and to simplify expressions for limit calculations.

Derivative of Sine Function

If you're diving into the nitty-gritty of calculus, you'll encounter derivatives early on. The derivative represents the rate at which a function is changing at any given point, and it's notated as \(f'(x)\) or \(\frac{d}{dx}f(x)\).

When it comes to the sine function, its derivative is beautifully straightforward:

The derivative of \(\sin(x)\) is \(\cos(x)\).

So, if you have a function like \(f(x) = \sin(kx)\), where 'k' is a constant, the derivative would be \(f'(x) = k\cos(kx)\).

This rule applies regardless of the value of 'k', and it's central to solving problems involving limits with L'Hôpital's Rule when you're faced with a sinusoidal function. Armed with this knowledge, you can differentiate, simplify, and find limits of more complex trigonometric expressions.