1. Evaluate the following limits using algebraic methods. If the limit does not exist, explain why, using the definition

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1. Evaluate the following limits using algebraic methods. If thelimit does not exist, explain why, using the definition of thelimit.
lim 𝑥3−1/𝑥3+5 when 𝑥→∞ lim 5𝑥3−8/4𝑥2+5𝑥 when𝑥→∞
2.Determine: (a) the derivative of each function using the limitdefinition of the derivative; (b) the instantaneous rate of changeof the function at the given value; (c) the tangent line that goesthrough the given x-value.
a. 𝑓(𝑥)=5𝑥2 +6𝑥−11;𝑥=−2
b. 𝑝(𝑞)=2𝑞2−4𝑞+5;𝑞=2
3. Suppose the productivity of a worker (in units per hour)after x hours of training and time on the
70𝑥2 job is given by: 𝑃(𝑥) = 3 + 𝑥2+1000.
Find and interpret 𝑃(20).
Find and interpret 𝑃′(20).
4.
𝑥2
14) A small business has weekly costs of 𝐶(𝑥) = 100 + 30𝑥 + 10,where x is the number of units
produced each week. The competitive market price for thisbusiness’ product is $46 per unit. Find the marginal profit.