Problem 34
Question
The function \(f(x)=\left\\{\begin{array}{cc}1-2 x+3 x^{2}-4 x^{3}+\ldots \text { to } & \infty, x \neq-1 \\ 1 & , x=-1\end{array}\right.\) is (A) continuous and derivable at \(x=-1\) (B) neither continuous nor derivable at \(x=-1\) (C) continuous but not derivable at \(x=-1\) (D) None of these
Step-by-Step Solution
Verified Answer
The function is neither continuous nor derivable at \(x = -1\).
1Step 1: Identify the Function
The function is defined piecewise. For \(x eq -1\), it follows the pattern \(1 - 2x + 3x^2 - 4x^3 + \ldots\). At \(x = -1\), \(f(x) = 1\).
2Step 2: Check Continuity
A function is continuous at a point \(x = a\) if \(\lim_{{x \to a}} f(x) = f(a)\). Here, \(x = -1\) gives \(f(-1) = 1\). We need the limit as \(x\) approaches \(-1\) from both sides to also equal 1.
3Step 3: Analyze the Series
Replacing \(x = -1\) in the series, we get \(1 - 2(-1) + 3(-1)^2 - 4(-1)^3 + \ldots\). This simplifies to an alternating series \(1 + 2 + 3 + 4 + \ldots\), which is a divergent series. The sum of this series doesn’t equal 1.
4Step 4: Conclusion on Continuity
Since the limit as \(x\) approaches \(-1\) from both sides doesn’t equal \(f(-1)\), the function is not continuous at \(x = -1\).
5Step 5: Check Differentiability
Differentiability implies continuity, so since \(f(x)\) is not continuous at \(x = -1\), it cannot be differentiable there.
Key Concepts
Differentiability of FunctionsPiecewise FunctionsDivergent Series
Differentiability of Functions
Differentiability is a property of a function that means we can find a derivative at a certain point. The derivative represents the rate at which the function is changing at that point. For a function to be differentiable at a particular point, it must first be continuous at that point. This is because differentiability implies continuity. It is like having a smooth curve without any sharp edges or jumps, ensuring that you can draw a tangent line at that point.
In simpler terms, imagine sliding smoothly on a curve without getting stuck or jumping. If there is a hole or a sharp corner, that means the function is not continuous, and thus not differentiable. In our problem, the function is not continuous at the point where it is evaluated, hence it cannot be differentiable at that point.
In simpler terms, imagine sliding smoothly on a curve without getting stuck or jumping. If there is a hole or a sharp corner, that means the function is not continuous, and thus not differentiable. In our problem, the function is not continuous at the point where it is evaluated, hence it cannot be differentiable at that point.
- If a function has a break, jump, or point of discontinuity at a point, it cannot be differentiable there.
- A continuous function can be differentiable, but not always. All differentiable functions must be continuous.
- Check for continuity before checking for differentiability.
Piecewise Functions
Piecewise functions are functions that have different expressions or "pieces" for different ranges of the input variable, often causing changes in behavior at certain points. These functions are defined by different formulas over various intervals. This makes them adaptable to model situations where the relationship between variables changes at specific thresholds.
In the given exercise, the function is defined in two parts: one expression for when the input is not -1, and a single constant value when the input is -1. This non-standard definition at specific points can sometimes create challenges in determining continuity and differentiability.
In the given exercise, the function is defined in two parts: one expression for when the input is not -1, and a single constant value when the input is -1. This non-standard definition at specific points can sometimes create challenges in determining continuity and differentiability.
- Each "piece" of a piecewise function applies to a distinct interval in its domain.
- Continuity and differentiability usually need to be checked at the boundaries where the pieces meet.
- Special attention should be given to the points where the definition of the function changes.
Divergent Series
A divergent series is a series whose terms do not converge to a limit, meaning they ebb out towards infinity or any undefined behavior. When trying to calculate sums of such series, rather than getting a finite number, they tend to grow unbounded without any convergence.
In this context, the series in the given function diverges as it simplifies to an infinite sum: 1 + 2 + 3 + 4 + ..., which means it does not settle to a particular finite value. This is key because, for continuity, the limit must exist and be finite, equivalent to the function value at a particular point.
In this context, the series in the given function diverges as it simplifies to an infinite sum: 1 + 2 + 3 + 4 + ..., which means it does not settle to a particular finite value. This is key because, for continuity, the limit must exist and be finite, equivalent to the function value at a particular point.
- If a series diverges, it means its sum cannot be computed to a specific number.
- Continuity at a point requiring the limit of the series to equal the function's value at that point is disrupted by divergence.
- Understanding the behavior of series can help in analyzing functions especially at points of interest.
Other exercises in this chapter
Problem 32
If \(f\) is differentiable function satisfying \(f(0)=0\), and if \(g(x)=\frac{f(x)}{x}\), then the value, that should be assigned to \(g(0)\), so that \(g\) is
View solution Problem 33
Let \(f(x)=\frac{1}{[\sin x]},[.]\) being the greatest integer function, then (A) \(f(x)\) is not continuous, where \(x \in(2 n \pi, 2 n \pi+\pi), n \in I\) (B)
View solution Problem 35
If \(f(x)=x+\frac{x}{1+x}+\frac{x}{(1+x)^{2}}+\ldots\) to \(\infty\), then at \(x=0\), \(f(x)\) (A) has no limit (B) is discontinuous (C) is continuous but not
View solution Problem 36
The function \(f(x)=\left[x^{2}\right]+[-x]^{2}\), where [.] denotes the greatest integer function, is (A) continuous and derivable at \(x=2\) (B) neither conti
View solution