Let alpha and beta be the distinct roots of ax^2 + bx + c = 0, then lim (x-alpha) (1 - cos(ax^2 + b

Q: Can I use L'Hopital's Rule here? It seems faster.

L'Hopital is a valid tool, but it can get messy with the chain rule and product rule. The standard limit approach is more elegant and less prone to calculation errors. It helps you see the structure of the function rather than just grinding through derivatives.

Q: Why does the $(x - \beta)^2$ term stay in the final answer?

Great question! As $x \to \alpha$, the term $(x - \beta)$ approaches $(\alpha - \beta)$, which is just a constant. It doesn't vanish like $(x - \alpha)$ does, so it remains as part of the final value.

Q: What if the roots were equal? Would the answer change?

If $\alpha = \beta$, then $(\alpha - \beta) = 0$, and the limit would be 0. The problem specifies distinct roots, which ensures the result is non-zero and meaningful.

Question

Let $\alpha$ and $\beta$ be the distinct roots of $ax^2 + bx + c = 0$, then $\lim_{x 	o \alpha} \frac{1 - \cos(ax^2 + bx + c)}{(x - \alpha)^2}$ is equal to

Options:

(A) $\frac{a^2}{2} (\alpha - \beta)^2$

(B) 0

(C) $-\frac{a^2}{2} (\alpha - \beta)^2$

(D) $\frac{1}{2} (\alpha - \beta)^2$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine standing on the edge of a cliff, looking down at a parabola defined by $f(x) = ax^2 + bx + c$. This is a geometric entity that carves its path through the Cartesian plane.

We are told that $\alpha$ and $\beta$ are its distinct roots. This means the parabola intersects the x-axis at exactly two points: $\alpha$ and $\beta$.

By the **Factor Theorem**, we can rewrite our quadratic as:
$$f(x) = a(x - \alpha)(x - \beta)$$
This expression is the key to unlocking the limit.

### The Indeterminate Trap

We are tasked with evaluating the following limit:
$$\lim_{x 	o \alpha} \frac{1 - \cos(ax^2 + bx + c)}{(x - \alpha)^2}$$

As $x$ approaches $\alpha$, the quadratic $ax^2 + bx + c$ approaches zero. Consequently, the numerator becomes $1 - \cos(0) = 0$, and the denominator also approaches zero.

We are staring at a classic $\frac{0}{0}$ indeterminate form. However, trigonometry provides the necessary tools to resolve this.

### The Trigonometric Transformation

We utilize the powerful half-angle identity: $1 - \cos 	heta = 2 \sin^2(\frac{	heta}{2})$. Let us set $	heta = a(x - \alpha)(x - \beta)$.

Substituting this into our limit, the numerator transforms into $2 \sin^2(\frac{a(x - \alpha)(x - \beta)}{2})$. Our limit now takes the form:
$$\lim_{x 	o \alpha} \frac{2 \sin^2(\frac{a(x - \alpha)(x - \beta)}{2})}{(x - \alpha)^2}$$

We recall the standard limit $\lim_{u 	o 0} \frac{\sin u}{u} = 1$. To apply this, we must ensure the denominator matches the square of the argument inside the sine function.

### The Elegant Cancellation

We multiply and divide by the square of the argument, which is $(\frac{a(x - \alpha)(x - \beta)}{2})^2$. This allows us to isolate the standard limit, which evaluates to $1$.

We are left with the remaining algebraic terms:
$$\lim_{x 	o \alpha} \frac{2 \cdot [\frac{a(x - \alpha)(x - \beta)}{2}]^2}{(x - \alpha)^2}$$

Simplifying the expression, the $2$ in the numerator and the $4$ in the denominator result in a factor of $\frac{1}{2}$. The $(x - \alpha)^2$ terms cancel out perfectly, leaving us with:
$$\frac{a^2(x - \beta)^2}{2}$$

Finally, substituting $x = \alpha$, we arrive at the beautiful result:
$$\mathbf{\frac{a^2(\alpha - \beta)^2}{2}}$$

This is the power of calculus—taking a complex, indeterminate expression and, through the lens of geometry and identities, reducing it to a simple, elegant constant.

Animated Solution for Mathematics - Limits, Continuity and Differentiability: Let $α$ and $β$ be the distinct roots of $a x^{2} + b x + c = 0$ , then $lim_{x \to α} \frac{1 - c o s ( a x ^{2} + b x + c )}{( x - α ) ^{2}}$ is equal to

Select Answer:

Visualized Solution

Visualizing the Quadratic Curve

Factoring via the Factor Theorem

Analyzing the Limit Expression

Substituting the Factored Form

Applying Trigonometric Identity

Rewriting the Numerator

Recalling the Standard Limit

Algebraic Manipulation

Evaluating the Standard Limit Part

Simplifying the Remaining Expression

Canceling Common Factors

Final Substitution and Result

The Sigma Insight: Evaluation of Limits & L'Hopital's Rule

Analyzing the Setup

The Indeterminate Trap

The Trigonometric Transformation

The Elegant Cancellation

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