Animated Solution for Mathematics - Limits, Continuity and Differentiability: ${\lim }_{x\to 0}\frac{(1-\cos 2x)(3+\cos x)}{x\tan 4x}$ is equal to

Imagine standing at the edge of a mathematical cliff. You are looking at the function

and you want to know what happens as $x$ gets infinitely close to $0$.

If you simply plug in $x=0$, you get

This is the indeterminate form, a classic sign that the function has a removable singularity—a hole in the graph. But fear not, for this is where the real magic begins.

Trigonometric Identities

To solve this, we need to break the indeterminacy. The term $1-\cos 2x$ is a classic trap for the untrained eye.

We know from our trigonometric toolkit that $1-\cos 2\theta =2{\sin }^2\theta$. By applying this identity, we transform our numerator into $2{\sin }^{2}x(3+\cos x)$.

Suddenly, the expression looks much more manageable. We have replaced a subtraction with a multiplication, which is exactly what we need to cancel out the problematic $x$ terms in the denominator.

Now, we recall the golden rules of calculus:

Our expression currently stands as:

We have a ${\sin }^{2}x$ in the numerator, which needs an $x^2$ in the denominator. We have an $x$ in the denominator, so we need one more $x$. Similarly, we have a $\tan 4x$ in the denominator, which needs a $4x$ in the numerator.

This is where the algebraic dance begins. We multiply and divide by $x$ and $4$ to perfectly align our terms.

By rearranging, we get:

Look at the elegance of this structure! The term ${\left(\frac{\sin x}{x}\right)}^2$ approaches $1^2=1$. The term $\frac{4x}{\tan 4x}$ approaches $1$.

And the remaining term $\frac{3+\cos x}{4}$ approaches:

When we multiply these together with our constant $2$, we get $2\cdot 1\cdot 1\cdot 1=2$.

The complexity melts away, leaving us with a simple, beautiful integer. The final result is 2. This is the power of limits: taking a chaotic, undefined expression and finding the precise value it is destined to reach.