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 discontinuous if left-hand and right-hand limits mismatch at boundary transitions. Non-differentiability occurs at discontinuities or sharp turns.

Core Logic

Simplify the greatest integer component [x]$[x]$ by expanding over integer intervals:

Step 1: Testing Continuity Limits

Check continuity at structural boundaries: At x = 0$x = 0$: LHM = 0$\text{LHM} = 0$, RHM = 0$\text{RHM} = 0$ $\implies$ Continuous. At x = 1$x = 1$: LHM = 1$\text{LHM} = 1$, RHM = 3$\text{RHM} = 3$ $\implies$ Discontinuous. At x = 2$x = 2$: LHM = 4$\text{LHM} = 4$, RHM = 5$\text{RHM} = 5$ $\implies$ Discontinuous.

Thus, α = 2$\alpha = 2$ points of discontinuity (x in 1, 2$x \in \{1, 2\}$).

Step 2: Testing Differentiability Parameters

Discontinuities automatically introduce non-differentiability. Now check smooth corners at the remaining continuous transition x = 0$x = 0$:

Thus, β = 3$\beta = 3$ points of non-differentiability (x in 0, 1, 2$x \in \{0, 1, 2\}$).

Pattern Recognition

Discontinuities automatically break differentiability. Always count them first before checking derivatives at smooth corner points.

Chapter Mix

Class 12 Maths: Limits, Continuity and Differentiability

Related Formula

Continuity definition condition frame:

Core Logic

Evaluate Left-Hand Limit (LHL) using standard forms [cite: 1410]: LHL = x arrow 0^- (1+ax)1/x = e^a$$\text{LHL} = \lim_{x \rightarrow 0^-} (1+ax)^{1/x} = e^a$$ [cite: 1410]

Given baseline definition states f(0) = 1+b$f(0) = 1+b$ [cite: 1410].

For Right-Hand Limit (RHL) to be finite and valid, the numerator tracking towards 0 means the denominator must also balance towards 0 to avoid divergence [cite: 1411]: x arrow 0^+ [(x+c)1/3 - 2] = 0 c1/3 = 2 c = 8$$\lim_{x \rightarrow 0^+} \left[(x+c)^{1/3} - 2\right] = 0 \implies c^{1/3} = 2 \implies c = 8$$ [cite: 1414]

Step 1: Applying L'Hopital's rule to the RHL

With c=8$c=8$, evaluate RHL limit expressions using derivatives [cite: 1411]: RHL = x arrow 0^+ 12√(x+4)(1)/(3)(x+8)-2/3 = (1)/(2(2))(1)/(3)(8)-2/3 = ((1)/(4))/((1)/(3 · 4)) = (1)/(4) · 12 = 3$$\text{RHL} = \lim_{x \rightarrow 0^+} \frac{\frac{1}{2\sqrt{x+4}}}{\frac{1}{3}(x+8)^{-2/3}} = \frac{\frac{1}{2(2)}}{\frac{1}{3}(8)^{-2/3}} = \frac{\frac{1}{4}}{\frac{1}{3 \cdot 4}} = \frac{1}{4} \cdot 12 = 3$$ [cite: 1411, 1415]

Step 2: Equating limits for parameter solutions

Equate continuous criteria milestones together [cite: 1416]: e^a = 1 + b = 3$e^a = 1 + b = 3$ [cite: 1416] e^a = 3 b = 2$$e^a = 3 \implies b = 2$$ [cite: 1416]

Compute the ultimate target combination product configuration [cite: 1416]: e^a · b · c = 3 · 2 · 8 = 48$$e^a \cdot b \cdot c = 3 \cdot 2 \cdot 8 = 48$$ [cite: 1416]

Pattern Recognition

Determining missing root constants inside indeterminate fraction structures handles calculations swiftly before running formal limits.

Chapter Mix

Class 12 Mathematics: Limits, Continuity and Differentiability

Related Formula

For a limit of the form x → 0 [g(x)]h(x)$\lim_{x \to 0} [g(x)]^{h(x)}$ where g(x) → 1$g(x) \to 1$ and h(x) → ∞$h(x) \to \infty$, the limit evaluates to:

Core Logic

Given the limit expression:

Using the standard form as f(2)=1$f(2)=1$, this transforms to:

Recognizing the definition of the derivative f'(2) = x → 0 (f(2+x)-1)/(x)$f'(2) = \lim_{x \to 0} \frac{f(2+x)-1}{x}$:

Given f'(2) = 4$f'(2) = 4$:

Step 1: Finding Intersection points with x-axis

Substitute α = 12$\alpha = 12$ into the equation of the curve:

To find where it meets the x-axis, set y = 0$y = 0$:

Testing for rational roots, x = -1$x = -1$ is a root because (-1)³ - (-1)² - 5(-1) - 3 = -1 - 1 + 5 - 3 = 0$(-1)^3 - (-1)^2 - 5(-1) - 3 = -1 - 1 + 5 - 3 = 0$.

Step 2: Factoring the cubic polynomial

Dividing x³ - x² - 5x - 3$x^3 - x^2 - 5x - 3$ by (x+1)$(x+1)$ gives:

The roots are x = -1$x = -1$ (repeated root) and x = 3$x = 3$. Therefore, the distinct real values of x$x$ where the curve intersects the x-axis are -1$-1$ and 3$3$, meaning it meets the x-axis exactly 2$2$ times.

Pattern Recognition

A repeated root like (x+1)²$(x+1)^2$ means the curve is tangent to the x-axis at that point, but it still counts as a meeting point. Always count distinct real roots when determining the number of meeting points with the coordinate axes.

Chapter Mix

Class 12 Mathematics: Limits, Continuity and Differentiability Class 11 Mathematics: Theory of Equations

Related Formula

Standard Taylor expansions near zero:

Core Logic

Let x - 1 = h$x - 1 = h$, where h → 0⁺$h \to 0^{+}$. The expression transforms into:

Substitute the expansions into the numerator:

Step 1: Match Coefficients for Existence

For the limit to be finite, the coefficient of h$h$ must vanish:

Equating the h³$h^3$ term to the given limit value:

Step 2: Solve System of Equations

Subtract equation (1) from (2):

From (1), μ = 6 + 6 = 12$\mu = 6 + 6 = 12$.

Pattern Recognition

When dealing with indeterminate form limits involving mixed trigonometric expressions with a non-zero denominator power, polynomial substitution using Taylor series is much cleaner and less prone to differentiation tracking mistakes compared to multiple L'Hôpital cycles.

Chapter Mix

Class 12 Mathematics: Limits, Continuity and Differentiability