Given that lim x→1 f(x) = 4    lim x→1 g(x) = −5    lim x→1 h(x) = 0,find the limits, if they exist. (If an answer does not exist, enter DNE.)(a)    lim x→1 [f(x) + 5g(x)](b)    lim x→1 [g(x)]3(c)    lim x→1 f(x)(d)    lim x→1 5f(x)g(x)(e)    lim x→1 g(x)h(x)(f)    lim x→1 g(x)h(x)f(x)

Use the graphs of f and g in the accompanying figure tofind the limits that exist. If the limit does not exist, explainwhy.(a) limx → 2 [f(x) + g(x)]

If g(x) is continuous for all real numbers and g(3) = -1, g(4) = 2, which of the following are necessarily true?I. g(x) = 1 at least onceII. lim⁡𝑥→3.5𝑔(𝑥)=𝑔(3.5)III. lim⁡𝑥→3−𝑔(𝑥)=lim⁡𝑥→3+𝑔(𝑥)​ I. g(x) = 1 at least onceII.  x→3.5lim​ g(x)=g(3.5)III.  x→3−lim​ g(x)= x→3+lim​ g(x)​

f (x) ={ 1x−1 if x 6 = 1;0 if x = 1.(a) Does limx→5f (x) exist? If so, what is it? Try and establish thevalidity of your answer formally using an epsilon-delta argument.If it exists, does it equal f (5)? Is this function continuous at thepoint x = 5?(b) Does limx→1f (x) exist? If so, what is it? Try and establish thevalidity of your answer formally using an epsilon-delta argument.If it exists, does it equal f (1)? Is this function continuous at thepoint x = 1?(c) Is this function continuous?2. Consider the functionf (x) ={x2 if x 6 = 0;−50 if x = 0.2

Decide whether the limit exists. If it exists, find its value.lim f(x)x→1

Suppose f and g are continuous functions such that g(4) = 4 and lim x → 4 [3f(x) + f(x)g(x)] = 35. Find f(4).