2. a) Use the (ε, δ)-definition of the limit of a function at a point to show that lim as x→1 (3x^2 − 5) = −2 . b) Suppose h is continuous on R and h(q) = 0 for every rational number q. Does it follow that h(x) = 0 for all x ∈ R? Justify your answer!

Q: Let n be a positive integer. For which n are the two infinite one-sided limits limx→0±1/xn^n equal?

A:

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

A: Let  I  be an open interval containing  c , and let  f  be a function defined on  I , except…

Q: Let f(x)={−2x+2  x≠1               {2   x=1. What is f(1)? If f(1) does not exist, type: 99999 What…

A:

Q: estimate the limit numerically to two decimal places or state that the limit does not exist

A: Given: Plugging the limit directly, we get:

Q: 2) By using E-S definition of limit, show lim (x+y+xy)cos÷ cos 음-0 COS (xy)(0,0)

A: Let f(x)=x+y+xy cos1x cos1y

Q: 1) Show that the limit doesn't exists : ysinx 3) lim (x,y)→(0,0) x²+y2

A: To show that the limit of a multivariable function does not exist,…

Q: Use the definition of limits at infinity to prove the limit

A: Definition of limits helps to find the limit of the function. A value is said to be the limit of…

Q: Evaluate the limit with either L'Hôpital's rule or previously learned methods. 5t In(t) lim t- o 8 +…

A:

Q: use the definition of limits atinfinity to prove the limit. {lim x approaches to infinity  ( 2/…

A: The function is . The limiting value of the function is to be found out as tends to infinity. To…

Q: 2] Show that the limit does not exist by considering the limits as (x, y) → (0, 0) along the…

A:

Q: 2 Use the definition of continuity to show f (x) = is x+3 continuous at x=1. Whenever you find a…

A:

Q: and If lim f(x) = lim g(x) = +o and flx) =1 and gtxl = e, what is the limit of the division of !!…

A: As given limit is infinitely by infinity form so use L'hopital's rule and substitute the given…

Q: Let Зх — 2. if x + 2, f (x) = 5, if x = 2. (a) Find lim→2 f(x) (b) Prove your answer to Part (a)…

A:

Q: Let f(x) = 4x² + 6x + 2. According to the definition, f'(x) lim h→0 (Enter the limit in reduced…

A: Given f(x)=4x2+6x+2

Q: Show that the absolute-value function y = abs(x) = |x| is NOT differentiable at x = 0. (Hint: by…

A:

Q: Express the following limit as an integral. 7 lim Σ (5(at)* + 4x; - 6) Δr over 12,61

A:

Q: Evaluate the following limits (if exist): 1-cos(ax) (c) lim- a+0 (d) lim/Vxr'+x-Vx°-cx), where c is…

A: To evaluate the given limit we are using L- Hospital rule

Q: 2. Use L'Hospital's rule to evaluate the following limit. cosh(3x) – 1 lim x→0 x2

A:

Q: Let f(x) V4x + 6. According to the definition, f'(x) = lim (Enter the limit in reduced form)

A:

Q: Compute the limit as x approaches infinity of x^e-x

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: 1) Describe in your own words, how to determine the limit of f(x) as x approaches a given point and…

A: Hi! you have posted multiple questions. As per norms, we are answering the first question as nothing…

Q: evaluate the limit  limit as x approaches infinity of x+cos(x)/ x+1

A: To find,

Q: Show that the absolute-value function y = t(x) = |x − 5| is NOT differentiable at x = 5. (Hint: by…

A:

Q: Prove that a function f is continuous at a limit point p iff the limit as x goes to p is f(p).

A:

Q: Express the limit as a definite integral. lim xk Ax [-3, 2] 4-0 k 1

A: Given

Q: evaluate the limit of xsin(1/x) as x approaches zero using the squeeze theorem.

A:

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

A:

Q: Compute the limit as x goes to 0 of [tan(x) - sin(3x)] / (x^2)

A: limx→0[tan(x)-sin(3x)]x2=[tan(0)-sin(3(0))]02=00

Q: 9. lim 15 1 + (i/n)6

A:

Q: Use the e-8 definition of a limit at infinity to prove that lim = 0. %3D

A:

Q: Compute the limit as xà0 of f(x)=(sin(61x))/x, if possible.

A: To find:limx→ 0sin61xx

Q: Prove that lim 4x – 2 = 10 using the epsilon-delta definition of a limit.

A: Let, ε>0 is any number. and δ>0 is also a number. For the given limit, fx=4x-2a=3L=10 From ε-δ…

Q: 10. Prove that lim x2 = 4 using the epsilon-delta definition of a limit. %3D X-2

A: limx→2x2=4

Q: Use the definition of the limit of a function of two variables to verify the limit.

A: We have to verify the limit: limx, y→1, 0x=1 According to limit value, x, y→1, 0 therefore, x=1y=0

Q: 1) Show that the limit doesn't exists : ysinx 3) lim (x.y)-(0,0) x²+y2

A:

Q: 1. Prove that lim (10r – 7) = 3 using epsilon-delta definition of a limit.

A:

Q: 4. Use the e - 6 definition of limit to show that lim (2: +3i) = 2 + 5i. 2+1+1

A:

Q: Prove: the limit as x approaches 0 1-cos(x)/x=0

A:

Q: Express the following limit as a definite integral: lim 2

A:

Q: Use the definition of limits at infinity to prove the limit.

A:

Q: A function f has a removable discontinuity at x = c if limf(x) exists by f is not continuous at x =…

A: Here, the given function is: f(x)=x-2|x|-2 So, function is not defined at x=2,-2

Q: 2. Use e -d of the limit to prove that lim z= 4

A: Given below the detailed solution

Q: Use the definition of limits at infinity to prove the limit.

A: A function f has a limit at infinity if there exists a real number L such that for all , there…

Q: Find the limit: lim f(x) as x approaches 0 of Kx2 - tan2 (mx)cos(mx)/x2     (m not =0)

A: Consider

Q: Does the existence and value of the limit of a function f(x) as x approaches a ever depend on what…

A: Yes, existence and value of the limit of a function f(x) as x approaches a depends on what happens…

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

A:

Q: Express the limit as an integral. Š (5(x; )² – 3(x; )³)ax over [0, 2] lim

A:

Q: Find the limit. lim Cn (3+x)- In (l+x)]

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.