Suppose that f is decreasing on [a,b] Prove that that lim Rpf) exists and equals to

Q: (a) Evaluate the following limit using the Squeeze Theorem sin(x) lim

A: Given: limx→∞sinxx, limx→0sinxx and limx→0sin5xsin6x

Q: Prove that lim [f(r, y) + g(x, y)] = L, + L2 (K, y) (a, b) where f(x, y) approaches L, and g(x, y)…

A: Given : lim(x, y) → (a, b) f(x, y) = L1     ...(1)lim(x, y) → (a, b) g(x, y) = L2    ...(2)   To…

Q: Prove that ƒ is continuous at c if and only if

A: We want prove that f is continuous at c if and only if limh→0f(c + h) - f(c)  that is limh→0f(c + h)…

Q: Let f(x) =, where Lx] is the greatest integer function. Show that for x + 0, 69. Then use the…

A:

Q: Evaluate using the Squeeze Theorem.

A:

Q: Let f E C(0, 1). Suppose that lim f(x) and lim f(x) exist (and are finite). Show that • f(x) is…

A:

Q: 5 Suppose that f: A -R is a function defined on an open interval A, and a E A. (a) Show that if r…

A:

Q: Suppose that f has a one-sided limit at each point of [a,b]. Prove that f is bounded on [a,b]

A: According to the given information, let f be one sided limit at each point of [a,b]. It is required…

Q: Let f : [0, 0) –+ R be a monotone function. Prove that lim f(x) = L for some number %3D L if and…

A:

Q: Let f be a real function, defined everywhere on R. Prove that if lim f(x) = L, then also lim f(-x) =…

A: Here we will use ε-δ  definition of limit Let f(x) be a function defined on an open interval around…

Q: Let f(x) = min(x,x²), where min(a, b) is the minimum of a and b. Prove rigorously that lim f(x) = 1.

A:

Q: If {an} is increasing and not bounded above, then lim an ylgn ihi

A:

Q: The limit lim x · In cos エ→0 is an indeterminate form of type 0 ·o. Select one: O True O False

A: LetL=limx→∞ xlncos1xL=∞.ln(cos(1∞))L=∞.ln(cos(0))L=∞.ln(1))L=∞.0Which is undefine

Q: use the formal definition of limit to prove that the function has a continuous extension to the…

A:

Q: Show that lim,--zo f (z)g(z) = 0 if limz¬zo f (z) = 0 and if there exist a positive number M such…

A:

Q: use the formal definition of limit to prove that the function has a continuous extension to the…

A: Given function is fx=x2-1x+1 and point is x=-1

Q: Suppose fé Ra, b]. Prove that b- a lim b- f(x) dr. k=1

A:

Q: If is a pole of f(z) then the residue of fat zo is lim (z- z0)f(=).

A:

Q: Use the Squeeze Theorem to evaluate the limit: lim a cos(9/2)

A:

Q: Suppose that f has a one-sided limit at each point of fa, b. Prove that f is bounde

A: In mathematics, a function f defined on some set X with real or complex values is called bounded if…

Q: Prove that if the limit of f(x) as x approaches c exists, then the limit must be unique.

A:

Q: Show that has a continuous extension to x = 2, and find that extension.

A:

Q: 3. Prove lim(a – 5x + 6) O using the epsilon-delta definition of a limit.

A: Explanation of the answer is as follows

Q: Let f : (a, ∞)→ℝ. Prove: If the limit of f as x→∞ exists, then it is unique.

A:

Q: Suppose that f : R -R is differentiable at z = 0. (a) Prove that (t) lim,-o EA = f'(0) (b) Give an…

A:

Q: Suppose that f is decreasing on [a,b]. Prove that that b. lim Rpf) exists and equals to Ilp||+0

A:

Q: Use the e-d definition of the limit of a function to prove that lim = 3. I→3 I – 2

A: Definition used -             If  limx→af(x) =L  , Then given any ε>0 , There exists…

Q: Suppose that a, > 0 and lim a, = ∞. Prove that Sa, diverges.

A:

Q: Use the Squeeze Theorem to evaluate the limit lim f(x), if 2x 1< f(r) < 2 on [-1, 3].

A: Squeeeze Theorem: Suppose that g(x)≤f(x)≤h(x) for all x close to a but not equal to a. If…

Q: Consider the following. lim x In(x) (a) Describe the type of indeterminate form (if any) that is…

A:

Q: Let if r is rational, f(x)= 0 if x is irrational. Prove that limx→a f(x) does not exist for any a €…

A:

Q: Prove the following limit, where the Squeeze Theorem is used: lim, 0(xtan-1(x)) = 0

A: Let us try to understand concept of Squeeze theorem-

Q: Show directly from definitions that if f(x) and g(x) are real func- tions, bounded on [0,1], and so…

A: Given: fx and gx are real functions, bounded on [0, 1], and limx→0+fx=2 and limx→0+gx=3 We have to…

Q: Let f(x) = min(x, x2), where min(a, b) is the minimum of a and b. Prove rigorously that lim f(x) =…

A:

Q: Use the table of values of f to estimate the limit. 4) Let f(x) = x2 + 8x – 2, find lim f(x). х-2 4)…

A: topic - limits

Q: 10. Prove that the statement "If lim f(r) exists, then f is continuous at a" is false by using a…

A: See the solution below.

Q: Show directly from definitions that if f(r) and g(r) are real func- tions, bounded on (0.1), and so…

A: Here we have given that fx and gx are real functions, bounded on 0,1, so that limx→0+fx=2 and…

Q: Prove that ƒ is continuous at c if and only if lim h--&gt;0 ƒ(c + h) = ƒ(c).

A: If part:

Q: If {a,} is decreasing and not bounded below, then lim an ylgn ihi

A:

Q: Suppose there is a number B such that |f(x)/x| < B for all x # 0. Then prove that lim f(x) = 0. 1.…

A: Given that: |f(x)/x|≤B

Q: for all a E R \ {0}. Using the e-ð definition of the limit of a function, prove that lim Do not use…

A:

Q: Prove the following limit using the 8,€ definition of lim,→p f(x) = q: lim(z,v)→(0,0) (x² – 2y² + 1)…

A: To prove:: limx, y→0,0x2-2y2+1=1 Proof; For ε&gt;0, consider δ=ε2; As limit is x, y→0,0, then…

Q: 30. Suppose that f, f', and f" are positive on R. Prove that lim f(x) = 00.

A: Let f, f' and f'' are positive on R. We need to prove that limx→∞fx=∞ Consider, the interval [a,∞).…

Q: 4. Use the definition of a limit to prove that lim z = a* for any real number a.

A:

Q: Let ƒ: S→R  be an uniformly continuous function. Finish the proof by showing that the limit lim_(x…

A:

Q: use the formal definition of limit to prove that the function is continuous at c.  ƒ(x) = x2 - 7, c…

A:

Q: Use the definition of "limit of f at c" to prove the limit of lim, 3(x² – 5x + 1) = -5

A:

Q: Use limits to prove the following result of Theorem 2.5.4: Iff and g are continuous at a,then (f)(g)…

A:

Q: 3 Prove the limit At en nti exists does not or exist lim ay tisinas lim a) b)

A: