The usual ε − δ definition of limits, Definition. limx→a f(x) = L exactly when for...

The usual ε − δ definition of limits, Definition. limx→a f(x) = L exactly when for every ε > 0 there is a δ > 0 such that for any x with |x − a| < δ we are guaranteed to have |f(x) − L| < ε as well.

1. Use the ε − δ definition of limits to verify that limx→1 (−2x + 1) = −1. [2]

2. Use the definition of limits that you didn’t use in answering question 1 to verify that limx→2 (−x + 2) does not =1. [2]

3. Use either definition of limits above to verify that limx→3 (x^2− 5)= 2. [3] Hint: The choice of δ in 3 will probably require some slightly indirect reasoning. Pick some arbitrary smallish positive number for δ as a first cut. If it doesn’t do the job, but x is at least that close, you’ll have more information to help pin down the δ you really need. Note: The problems above are probably easiest done by hand, though Maple and its competitors do have tools for solving inequalities which could be useful.

5. Compute limx→0 sin (x + π)/x by hand. [1

Expert Solution

Colby Messinger answered 2 years ago

1. Use the ε-δ definition of continuity to prove that (a) f(x) = x 2 is...

1. Use the ε-δ definition of continuity to prove that (a) f(x) = x 2 is continuous at every x0. (b) f(x) = 1/x is continuous at every x0 not equal to 0. 3. Let f(x) = ( x, x ∈ Q 0, x /∈ Q (a) Prove that f is discontinuous at every x0 not equal to 0. (b) Is f continuous at x0 = 0 ? Give an answer and then prove it. 4. Let f and g...

Use the ε-δ definition of limits to prove that lim x3 −2x2 −2x−3 = −6. 3...

Use the ε-δ definition of limits to prove that lim x3 −2x2 −2x−3 = −6. 3 markx→1 Hint: This question needs students have a thorough understanding of the proof by the ε-δdefinition as well as some good knowledge of what is learnt in Math187/188 and in high school, such as long division, factorization, inequality and algebraic manipulations. End of questions.

Let f(x) and g(x) be two generic functions. Assume limx→0(f(x)+2g(x))=2 & limx→0(f(x)−g(x))=8. Compute limx→1(f(lnx)/g(x2−x)). A. It...

Let f(x) and g(x) be two generic functions. Assume limx→0(f(x)+2g(x))=2 & limx→0(f(x)−g(x))=8. Compute limx→1(f(lnx)/g(x2−x)). A. It cannot be computed B. 2 C. 4 D. -3 E. -4

2. Evaluate the following limits. (a) lim x→1+ ln(x) /ln(x − 1) (b) limx→2π x sin(x)...

2. Evaluate the following limits. (a) lim x→1+ ln(x) /ln(x − 1) (b) limx→2π x sin(x) + x ^2 − 4π^2/x − 2π (c) limx→0 sin^2 (3x)/x^2

Expand in Fourier series: f(x) = x|x|, -L<x<L, L>0 f(x) = cosx(sinx)^2 , -pi<x<pi f(x) =...

Expand in Fourier series: f(x) = x|x|, -L<x<L, L>0 f(x) = cosx(sinx)^2 , -pi<x<pi f(x) = (sinx)^3, -pi<x<pi

Use the definition f '(x) = lim tends to 0 (f(x+h) - f(x) / h) to...

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sketch the fourier series of f(x) on the interval -L <= x <= L for A)...

sketch the fourier series of f(x) on the interval -L <= x <= L for A) f(x) = (x when x < L/2 and 0 when > L/2) B) f(x) = e^-x

1)For the function f(x)=6x^2−2x, evaluate and simplify. f(x+h)−f(x)/h 2)Evaluate the limit: limx→−6 if x^2+5x−6/x+6 3)A bacteria...

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prove using the definition of derivative that if f(x) and g(x) is differentiable than (f'(x)g(x) -...

prove using the definition of derivative that if f(x) and g(x) is differentiable than (f'(x)g(x) - f(x)g'(x))/g^2(x)

For the function f(x) = x(x+1) find the exact formula for f′(x). Use only the definition...

For the function f(x) = x(x+1) find the exact formula for f′(x). Use only the definition of the derivative.

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