6. Consider a function f with the following property: if g is any function for which lim g(x) does not exist, then lim (f(r)+ g(x)) also does not exist. Use proof by contradiction to show that this happens if and only if lim f(x) does exist.

Q: 4 if x 2 Find the value of c such that lim f(x) exists. x→2

A: Given the piecewise function :

Q: 1. Let f(x) = 3² – 2x + 1. Compute lim (x)f – (4 + x)f 0+-4

A:

Q: 3. Explain why the following limits do not exist. 1 (a) lim x-5 (x-5)2 -x² +3, (b) lim h(x), where…

A: (a) limx→51(x-5)2 Let  f(x)=1(x-5)2 As x approaches 5, (x-5)2 approaches 0, and so f(x) approaches…

Q: Sketch the graph of the f(x) function and use it to determine the values of a for which lim f(x)…

A: Definition of limit at a point of any function : Suppose we have a function defined on real number .…

Q: (4) Let f(x) be the function whose domain is all of R given by the rule 42 if r € Z and f(x) = 10…

A:

Q: _15. What is lim f(x)? X-4 A. 0 В.1 С.2 D. Does Not Exist

A: Given: Following graph is given. To Find: 15. limx→4fx=?

Q: 6. Is it true that the limit of a function lim f(c) = L is the same as evaluating a function f(c)?

A: the limit of a function lim f(c) = L is the same as evaluating a function f(c)

Q: Let f(x) be a piecewise function defined by x+1 if x 2 Find lim f(x). A) 3 (в) 5 c) 2 (D) the limit…

A: Given : Piecewise function fx = x + 1if x<05if 0≤x< 2 x2 - 1if x>2   To find : limx→1+fx

Q: 11. Use the definition of the limit of a function to prove that lim g(x) =- 6 if %3D (16x²-9 4x +3 3…

A:

Q: Prove that if lim, „a g(x) = 0 and lim, „«f(x) exists and is not 0, then %3D f(x) lim does not exist…

A:

Q: If g(x) = |x|, then lim g(x) exists. True False

A: Given  g(x) = x We have to check if limx→0 g(x) exists or not.

Q: - Prove that if limf(x) = L, then lim |f(x)| = |L|.

A: Given:              limx→c f(x) = L To prove:               limx→c |f(x)| =|L|

Q: Show that the Product Law cannot be applied to lim f(x)g(x) if f(x) = x and g(x) =x-.

A: We have to find the limit of the function :                 limit as x approaches to zero ;…

Q: 3. is a type of discontinuity when the lim and lim f(x) = M where L# M and either f(a) does not…

A: Discontinuity 1: When the limx -> a- f(x) = L and limx -> a- f(x) = M, where L ≠ M and either…

Q: 3. Evaluate the limit of lim(x² + x + Vx + 1) x→3

A: A limit of a function f(x) is defined as a value, where the function reaches as the limit reaches…

Q: If f is an odd function and if lim f(x) exists Prove that this limit must be zero.

A: if limit is exist then right hand limit and left hand limit must be equal

Q: Prove that lim x approaches c  .......... ƒ(x) = L if and only if lim hS0 ƒ(h + c) = L.

A: We have to prove that if limh→0fh+c=L then limx→cfx=L. First, consider limh→0fh+c=L and solve the…

Q: 02 3 What is lim f(x) for the function f, defined as f(x)= x-1 Does not exist X-1 √x-1 x²+x-2 X-1 if…

A: The given function is  fx=x-1x-1if x<1x2+x-2x-1if x>1 we find the left hand limit at x=1

Q: Show that if lim f(x) = lim h(x) = 3, and if f(x) ≤ g(x) ≤h(x) for 2-0 x-0 all z € (-1,1), then lim…

A: Given that limx→0fx=limx→0hx=3 and fx≤gx≤hx for all x∈−1,1. We have to prove that limx→0gx=3.…

Q: Sketch the graph of the f(x) function and use it to determine the values of a for which lim f (x)…

A:

Q: 3].. Use € Use e – 8 definition of a limit of function to show that: 2x = 1. x→1 1 + x a) lim X- b)…

A:

Q: what is the lim f(x)9(x) if lim g(x) ln f(x) = +∞ x→a x→a

A:

Q: 4.) Consider the function defined as Find the following limits. a.) lim f(x) b.) lim f(x) c.) lim…

A:

Q: If the limit of two different functions, lim f(x) and lim g(x) XC exist, then so does lim[f(x) -…

A: for example limx->2f(x)= 5limx->2g(x)=∞ f(x) and g(x) exists now find the limit f(x)-g(x) when…

Q: 1. Determine the lim f(x) if f(x) : = {x*. if #2 lo, if x = 2 2. Find lim f(x) if f (x) = 2*

A: (1) Given that, f(x) = x2 ; if x is not equal to 2 f(x) = 0 ; if x = 2 Then we get, As x -> 2,…

Q: 4. The graph of a function f(x) is shown at right. If g(x) = V1+ 3x , find lim [f(x) + 3 g(x)]. %3D…

A: The graph of the function fx is given. From that graph it is clear that, the function fx, at x=1…

Q: 11. Determine the limits. In(x) a. lim X→1 X-1

A:

Q: Prove that lim ƒ(x) = L if and only if lim ƒ(h + c) = L.                   X->c…

A: Given : The statement,  limx→cf(x) = L if and only if limh→0f(h+c) = L

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: use the definition of differentiability f'(x) = lim- h-0 f(x+h)-f(x) h to For f(x) = %3D 6. (x-1)…

A: fx=x2x-1 Now,…

Q: A- The result of the following limit is 2x² + 3x lim x→∞0x³ + x + 1

A:

Q: If a function f is not defined at x = c, then limf(x) does not exist. True False

A: Limit of a function at a point exist mean left hand side limit and right hand side limit at that…

Q: 2. Suppose f is a positive function. If lim,»a f(x) = 0 and lim, a g(x) = 0, show that lim [f(x)]•w…

A: limx→afx=0  &  limx→agx=∞

Q: 14. Evaluate the following limits. a) lim (x² + 2x – 3) X-0 b) lim (x3 – 2x² + 3x + 4) X-

A:

Q: Show that the Product Law cannot be applied to lim f(x)g (x) if f(x) = x and g(x) = x- 1. x→ 0

A: We have to find the limit of the function: limit as x approaches to zero;  f(x)g(x) = x[x^{-1}] =…

Q: Determine the value for c so that lim f(x) exists. X-3 f(x) = 3**c, for x3 %3D

A: NOTE: Refresh your page if you can't see any equations. . find the left side limit of the function…

Q: Determine the value for c so that lim f(x) exists. X-3 f(x) = +c, for x3 The value of c is.

A: Topic = Limits

Q: 1. Give an example of a function g for which lim g(x) does not exist, but the function I+2021 value…

A: We have to find the required example first.

Q: It was determined that the lim f(x) = 0 and f(2) = 0. Can it be concluded, definitively, that f'(2)…

A:

Q: (2) Suppose that a function f(x) is defined for all real values x except x* x,, then lim f(x) does…

A: For limit to be exist at a point, left hand side and right hand side values of function should be…

Q: Let f(x)=-4x-2 and g(x)=x'. Find the limit. lim g(flx)) x-3

A: Given that  f(x) = - 4x - 2 g(x) = x3

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

A: Given that f is continuous on (0,1)

Q: Suppose that f (x) is a function such that lim f(r) and lim f (r) both exist. Does this mean lim…

A:

Q: 12 Show that the following limits do not exist. 1 lim X0x (a) (x > 0), 4,

A:

Q: 1- Evaluate the following limits, if they exist. If they do not exist, state why.! lim f(x), where…

A: Given the query is to find the limit of the expression.

Q: 4. Sketch a function f such that: lim f(x) and f(4) exist and are equal to one another.

A: We need to sketch a function for which limx→4f(x) and f(4) exist and equal to one another.

Q: 10. Use the squeeze theorem to find lim f(x) if 2x+2 ≤ f(x) ≤ x² +3. x-1

A: Given: Use the squeeze theorem to find limx→1f(x) if 2x+2≤f(x)≤x2+3

Q: Prove rigorously that lim (x2 – x) = 6. x→3

A: We have to prove  limx→3x2-x=6 Now let  L=limx→3x2-x

Q: Suppose that lim, →o(f(x) – 9(x)) = 0 and h(x) is a function with domain R. a) Assume that |h(x) –…

A: We will answer the first question as we only answer one question at a time. Please resubmit the…

Q: Let f(x) be the function defined by x²+ 4x (xs3) 2x - 1 (x>3) the rule f(x) Find the one-sided limit…

A:

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.