MAT-120
Assignment – I
Sec: 07 Total: 50
Please answer all the questions:
1. For the function f, whose graph is shown below state the following:
a. lim 𝑓(𝑥) b. lim 𝑓(𝑥) c. Equation of Vertical asymptotes.
𝑥→6 𝑥→−3
d. Discuss where the following function is discontinuous and why?
𝑥+4−3
2. a. Find limit lim (𝑥√2−6𝑥+5)
𝑥→5
5𝑥 + 24 𝑖𝑓 𝑥 < −3
b. If 𝑓(𝑥) = { 𝑥 2 𝑖𝑓 − 3 ≤ 𝑥 < 4
1 − 2𝑥 𝑖𝑓 𝑥 ≥ 4
Evaluate the following limits if they exists
i. lim 𝑓(𝑥) ii. lim 𝑓(𝑥) iii. lim 𝑓(𝑥)
𝑥→−3 𝑥→4 𝑥→12
1
3. Find limits:
3 2+3𝑥−5𝑥 2 3𝑥 2 −𝑥−2
a. lim √ b. lim √9𝑥 2 + 𝑥 − 3𝑥 c. lim
𝑥→+ ∞ 1+8𝑥 2 𝑥→+∞ 𝑥→∞ 5𝑥 2 +4𝑥+1
𝑒 sin 𝑥
4. a. Find the values of 𝑥, if any, at which 𝑔(𝑥) = 8−√𝑥2−36 is continuous.
𝑥 2 −4
𝑖𝑓 𝑥 < 2
𝑥−2
b. A function is defined as 𝑓(𝑥) = { 𝑎𝑥 − 𝑏𝑥 + 3 2
𝑖𝑓 2 ≤ 𝑥 < 3
2𝑥 − 𝑎 + 𝑏 𝑖𝑓 𝑥 ≥ 3
find constants 𝑎 and 𝑏 such that the function is continuous everywhere.
𝑥 2 + 5 𝑖𝑓 𝑥 ≤ 1
c. b. Show that 𝑓(𝑥) = { is continuous but not differentiable
𝑥 + 10 𝑖𝑓 𝑥 > 1
at 𝑥 = 1. Sketch the graph of 𝑓(𝑥).
𝑥+3
5. a. Find all values of x at which the tangent line to the given curve 𝑦 = 𝑥+2 is
perpendicular to the line 𝑦 = 𝑥.
𝑑𝑦 𝑥−5
b. Find 𝑑𝑥 | if 𝑦 = (2𝑥 7 − 𝑥 2 ) (𝑥+5).
𝑥=2
𝑑𝑦
c. Find 𝑑𝑥 using implicit differentiation if tan−1(𝑥 2 𝑦) = 𝑥 + 𝑥𝑦 2 .
