Calculus Limits and Continuity

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Questions and Answers

What is the limit of the function x - |x| as x approaches 0 from the right?

0 (correct)

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-1

What does it imply if lim (x→0+) of the function x - |x| is not equal to lim (x→0−) of the same function?

The function has a vertical asymptote at x = 0.

The limit exists at x = 0.

The limit does not exist at x = 0. (correct)

The function is continuous at x = 0.

What value does f(x) approach as x approaches 1/2 for rational x?

1/4

1/2 (correct)

Which statement best describes the function f defined as f(x) = x if x is rational and f(x) = 1 - x if x is irrational?

It is only continuous at x = 1/2. Signup and view all the answers

For the limit lim (x→∞) of f(x) = x/(1 + sin(x)), what can be concluded about its existence?

The limit does not exist. Signup and view all the answers

What is the limit of f(x) as n approaches infinity where f(x) = x(1 + sin(x))?

∞ Signup and view all the answers

How can the expression |f(x) - f(1/2)| be interpreted to establish continuity of f at x = 1/2?

The expression indicates that f(x) gets arbitrarily close to f(1/2) as x approaches 1/2. Signup and view all the answers

If lim (x→∞) f(x) does not exist, what conclusion can we draw about the behavior of the function?

The function oscillates infinitely. Signup and view all the answers

What condition must be satisfied for the series to converge based on the ratio test?

$x^2 < 1$ Signup and view all the answers

At which points does the series diverge?

-1 Signup and view all the answers

What is the maximum interval of convergence for the Taylor series of the given function?

Signup and view all the answers

Is the function f(x) = x cos(x) differentiable at x = 0?

No, it is not differentiable at x = 0 because the limit does not exist. Signup and view all the answers

What is the value of f(0) for the function f(x) defined as x cos(x) if x ≠ 0 and 0 if x = 0?

0 Signup and view all the answers

For the function f(x) = x^3 sin(x) if x ≠ 0 and 0 if x = 0, what happens to f'(x) as x approaches 0?

It approaches 0. Signup and view all the answers

What is the continuity status of f'(x) at x = 0 for the function f(x) = x^2 cos(x) if x ≠ 0 and 0 if x = 0?

f'(x) is continuous at x = 0. Signup and view all the answers

Which limit is used to determine the differentiability of f(x) = x cos(x) at x = 0?

lim (f(x) - f(0))/(x - 0) Signup and view all the answers

What is the radius of convergence for the Taylor series of f(x) = e^x sinh(x)?

∞ Signup and view all the answers

Which sequence is used in the limit argument for differentiability of f(x) at x = 0?

xn = nπ Signup and view all the answers

What is the formula for the n-th derivative of f(x) = e^x sinh(x) evaluated at 0?

2n-1 Signup and view all the answers

Why does the limit lim (cos(xn)) as xn approaches 0 not exist for xn = nπ?

Because cos(xn) oscillates between -1 and 1. Signup and view all the answers

What does the expression f(x) = x^2 ln|x| signify about the behavior of f at x = 0?

f is not defined at x = 0. Signup and view all the answers

What does the ratio test show for the series derived from the function f(x) = x^2n+1/(2n+1)!?

The series converges for all x Signup and view all the answers

Which series represents the Taylor series expansion of f(x) = e^x sinh(x) around 0?

∑ (2n-1) x^n/n! Signup and view all the answers

For the function f(x) = x sin(x), what is the correct expression for the first derivative?

cos(x) + x sin(x) Signup and view all the answers

What can be concluded from the expression an = (2n-1)/n! in terms of convergence?

The limit approaches zero, indicating convergence for all x. Signup and view all the answers

What does the n-th term of the Taylor series for the function f(x) = x^2n+1/(2n+1)! include?

x^(2n+1) / (2n+1)! Signup and view all the answers

Which of the following represents a mistake in finding the radius of convergence?

Assuming an is increasing without checking limits. Signup and view all the answers

What is the necessary condition for the convergence of the series $\sum_{n=0}^\infty \frac{1}{n^p}$?

$p > 1$ Signup and view all the answers

In proving that the series $\rac{2n}{(2n + 1)n}$ is convergent, which test is often employed?

Comparison test Signup and view all the answers

For which values of $x$ does the series $\rac{1}{2n} \sum_{n=0}^{\infty} \frac{2n x^{2n+1}}{2n(2n + 1)}$ converge?

$[-1, 1]$ Signup and view all the answers

What is the derivative of the function $sinh^{-1}(x)$?

$\frac{1}{\sqrt{1 + x^2}}$ Signup and view all the answers

Which of the following reflects the relationship between $\sum_{n=0}^{\infty} \frac{1}{(2n + 1)(3n + 1)}$ and convergence?

It converges by the limit comparison with $\sum_{n=0}^{\infty} \frac{1}{n^2}$. Signup and view all the answers

What is indicated by the inequality $\frac{42^n}{22^n(2n + 1)3n + 1} \le \frac{1}{\sqrt{n}}$?

The series converges for all n. Signup and view all the answers

In the context of the series development, what is the main purpose of using induction?

To establish a base case and inductive step for convergence. Signup and view all the answers

How is the function $sinh^{-1}(x)$ formally defined in terms of its derivative?

$sinh^{-1}(x) = \int \frac{1}{\sqrt{1 + x^2}}$ Signup and view all the answers

What is the condition for divergence of the series based on the percentage of x?

The series diverges if $x^2 > 1$ Signup and view all the answers

What is the radius of convergence for the Taylor series of the given function?

1 Signup and view all the answers

At which points does the series diverge according to the analysis?

At $x = -1$ and $x = 1$ Signup and view all the answers

What does the limit of the term $b_{n+1}/b_n$ approach as n approaches infinity?

$x^2$ Signup and view all the answers

What is the form of the terms in the series represented by $b_n$?

$(-1)n-1 n x^{2n-2}$ Signup and view all the answers

Study Notes

Limits

The limit of (x - |x|)/x as x approaches 0 does not exist because the limit from the right is 0 and the limit from the left is 2.
The limit of x(1 + sin(x)) as x approaches infinity does not exist because for the sequence xn = 2nπ + 3π/2, the limit of f(xn) is 1, but for the sequence yn = nπ, the limit of f(yn) is infinity.

Continuity

The function f(x) defined as x for rational x and 1 - x for irrational x is continuous at x = 1/2.
The same function, f(x), is not differentiable at x = 0 because the limit of (f(x) - f(0))/x as x approaches 0 from the left does not exist.

Differentiability

The function f(x) defined as x cos(x) for x ≠ 0 and 0 for x = 0 is not differentiable at x = 0. This is because the limit of (f(x) - f(0))/x as x approaches 0 does not exist, as demonstrated by the sequence xn = 1/nπ.

Taylor Series

The Taylor series of f(x) = x³sin(x) around 0 is ∑(n=0 to ∞) x^(2n+1)/(n!)(2n+1)!.
The Taylor series of f(x) = e^x sinh(x) around 0 is ∑(n=1 to ∞) 2^(n-1) x^n / (n!)².
The Taylor series of f(x) = x sin(x) around 0 is ∑(n=0 to ∞) (-1)^n x^(2n+1) / (2n+1)!.

Convergence of Series

The Taylor series of f(x) = x³sin(x) around 0 converges for all x ∈ R.
The Taylor series of f(x) = e^x sinh(x) around 0 converges for all x ∈ R, as demonstrated by the ratio test.
The maximum interval of convergence for the Taylor series of f(x) = x sin(x) around 0 is [-1, 1], as demonstrated by the ratio test and convergence at x = ±1.

Taylor Series of Inverse Functions

The Taylor series of f(x) = sinh⁻¹(x) around 0 is ∑(n=0 to ∞) (-1)^n (2n)! x^(2n+1) / (2^(2n)(n!)²(2n+1)).
The radius of convergence for the Taylor series of sinh⁻¹(x) is 1.
The maximum interval of convergence for the Taylor series of sinh⁻¹(x) is (-1, 1).

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Description

This quiz covers key concepts in calculus, including limits, continuity, and differentiability of functions. Understand the behavior of functions as they approach certain points and their performance under various conditions. Additionally, explore the Taylor series expansion for specific functions.