If t→ 0(∫₀¹ (3x + 5)^t dx)(1)/(t) = (α)/(5e)((8)/(5))(2)/(3)$\lim_{t\to 0}\left(\int_0^1 (3x + 5)^t dx\right)^{\frac{1}{t}} = \frac{\alpha}{5e}\left(\frac{8}{5}\right)^{\frac{2}{3}}$, then α$\alpha$ is equal to

Related Formula

A function is non-differentiable at sharp corner transition points where left-hand and right-hand derivatives do not match.

Core Logic

Analyze the behavior of powers of x$x$ across significant transition domains: For x < -1$x < -1$: x$x$ is the largest because higher odd powers of negative fractions decrease rapidly (x > x³ > x⁵...$x > x^3 > x^5...$). For -1 ≤ x < 0$-1 \le x < 0$: x²¹$x^{21}$ is largest (closest to zero from below). For 0 ≤ x < 1$0 \le x < 1$: x$x$ is largest. For x ≥ 1$x \ge 1$: x²¹$x^{21}$ is largest.

Step 1: Continuity and Differentiability Checks

At critical intersection boundaries x = -1, 0, 1$x = -1, 0, 1$, f(x)$f(x)$ matches continuous values perfectly, so n = 0$n = 0$. Now check derivative transitions f'(x)$f'(x)$:

At x = -1$x = -1$: LHD = 1$\text{LHD} = 1$, RHD = 21(-1)²⁰ = 21 Non-differentiable$\text{RHD} = 21(-1)^{20} = 21 \implies \text{Non-differentiable}$. At x = 0$x = 0$: LHD = 0$\text{LHD} = 0$, RHD = 1 Non-differentiable$\text{RHD} = 1 \implies \text{Non-differentiable}$. At x = 1$x = 1$: LHD = 1$\text{LHD} = 1$, RHD = 21(1)²⁰ = 21 Non-differentiable$\text{RHD} = 21(1)^{20} = 21 \implies \text{Non-differentiable}$.

Step 2: Conclusion

Thus, the function is non-differentiable at exactly 3 points (x = -1, 0, 1$x = -1, 0, 1$), so m = 3$m = 3$. Since n = 0$n = 0$:

Pattern Recognition

Maximum boundary tracking curves for standard power elements always form continuous shapes but introduce non-differentiable sharp corners at every intersection crossover point.

Chapter Mix

Class 12 Mathematics: Limits, Continuity and Differentiability

Related Formula

For a finite limit x→ 0 (f(x))/(xⁿ) = L$\lim_{x\to 0} \frac{f(x)}{x^n} = L$, the lowest powers of x$x$ below degree n$n$ in the polynomial f(x)$f(x)$ must vanish.

Core Logic

Let the 4th-degree polynomial be:

Given:

For the limit to exist and equal 5, the terms dx$dx$ and e$e$ must be 0$0$, and the coefficient of x²$x^2$ must be equal to 5:

Thus, the polynomial simplifies to:

Step 1: Use Extrema Conditions

Differentiating f(x)$f(x)$ with respect to x$x$:

Since f(x)$f(x)$ has extreme values at x=4$x=4$ and x=5$x=5$, f'(4) = 0$f'(4) = 0$ and f'(5) = 0$f'(5) = 0$. This means 4$4$ and 5$5$ are roots of the quadratic factor 4ax² + 3bx + 10 = 0$4ax^2 + 3bx + 10 = 0$.

Step 2: Solve Coefficients

Using properties of roots for 4ax² + 3bx + 10 = 0$4ax^2 + 3bx + 10 = 0$:

Substituting 4a = (1)/(2)$4a = \frac{1}{2}$:

Our full polynomial is:

Step 3: Calculate f(2)

Evaluate at x = 2$x = 2$:

Pattern Recognition

Whenever a limit explicitly matches a denominator power xⁿ$x^n$, it directly yields both the lower-order coefficients as zeroes and the xⁿ$x^n$ coefficient as the limit value.

Chapter Mix

Class 12 Mathematics: Limits, Continuity and Differentiability

Related Formula

For continuity at x=0$x=0$, f(0) = x → 0 f(x)$f(0) = \lim_{x \to 0} f(x)$.

Core Logic

We need to evaluate the limit:

Adding and subtracting x$\tan x$ inside the numerator:

Divide individual parts by x³$x^3$ across standard series layouts directly yields the combined fractional evaluation equal to 2.

Step 1: Final Resolution

The limit evaluates cleanly to 2. Therefore, for continuity, f(0) = 2$f(0) = 2$.

Pattern Recognition

Expansion of expansion functions like ( x)$\tan(\tan x)$ simplifies smoothly when paired strategically with basic structural Taylor series expansions.

Chapter Mix

Class 12 Mathematics: Limits, Continuity and Differentiability

Related Formula

Sum of roots for a quadratic equation Ax² + Bx + C = 0$Ax^2 + Bx + C = 0$ satisfies:

Core Logic

We need to find t → -1 (αₜ + βₜ) = a + b$\lim_{t \to -1} (\alpha_t + \beta_t) = a + b$:

Let y = t+2$y = t+2$. As t → -1$t \to -1$, y → 1$y \to 1$.

Step 1: Evaluate Limit

Applying L'Hopital's Rule or standard limit templates:

Squaring the sum alignment:

Pattern Recognition

Treating (α + β)$\lim(\alpha + \beta)$ collectively allows direct evaluation via standard root identities without solving for individual root entities.

Chapter Mix

Class 12 Mathematics: Limits, Continuity and Differentiability Class 11 Mathematics: Quadratic Equations

Related Formula

The greatest integer function [x]$[x]$ is discontinuous at all integer points. The absolute value function |x-x₀|$|x-x_0|$ is continuous everywhere but non-differentiable at its corner tip x = x₀$x = x_0$.

Core Logic

Break down the function f(x) = [x] + |x-2|$f(x) = [x] + |x-2|$ in the open domain (-2, 3)$(-2, 3)$ across sub-intervals between integers:

Step 1: Count Discontinuity Points (m$m$)

Evaluate the limits at internal integers -1, 0, 1, 2$\{-1, 0, 1, 2\}$:

• At x = -1$x = -1$: LHL = 1$\text{LHL} = 1$, RHL = 2 ⇒$\text{RHL} = 2 \Rightarrow$ Discontinuous.
• At x = 0$x = 0$: LHL = 1$\text{LHL} = 1$, RHL = 2 ⇒$\text{RHL} = 2 \Rightarrow$ Discontinuous.
• At x = 1$x = 1$: LHL = 1$\text{LHL} = 1$, RHL = 2 ⇒$\text{RHL} = 2 \Rightarrow$ Discontinuous.
• At x = 2$x = 2$: LHL = 1$\text{LHL} = 1$, RHL = 2 ⇒$\text{RHL} = 2 \Rightarrow$ Discontinuous.

Thus, f(x)$f(x)$ is discontinuous at exactly 4 integer locations , meaning m = 4$m = 4$.

Step 2: Count Non-Differentiability Points (n$n$)

Since discontinuity automatically implies non-differentiability, the points -1, 0, 1, 2$\{-1, 0, 1, 2\}$ are non-differentiable. Let's check if there are other sharp corners. The modulus part |x-2|$|x-2|$ turns sharp at x=2$x=2$, which is already covered in our discontinuity list. Hence, there are no additional non-differentiable points.

Thus, n = 4$n = 4$.

Step 3: Total Evaluation

Calculate the Σ$$\sum$$ requested :

Pattern Recognition

For expressions containing [x]$[x]$, the discontinuity at integers usually drives the overall non-differentiability tally, making any coincidental sharp points from continuous elements redundant if they happen at the exact same integers.

Chapter Mix

Class 12 Mathematics: Limits, Continuity and Differentiability