1) Let f(x) = e^(1/x), where x does not equal to 0. a) Describe the behavior of
ID: 3342434 • Letter: 1
Question
1) Let f(x) = e^(1/x), where x does not equal to 0.
a) Describe the behavior of f(x) as x--> infinity, and as x--> negative infinity; that is, find lim fx) as x app infinity and lim f(x) app negative infinity.
b) Describe the behavior of f(x) as x--> 0+ and as x--> 0-; that is, find lim f(x) as x-->0+ and lim f(x) as x--> 0-.
c) Find, if anym the asymptotes of f.
d) For what values of x is the function increasing? Decreasing? Neither?
e) For what values of x is the function concave up? Concave down? Neither?
f) Find, if any, the inflection points of f.
2) Let g(x) = ln(x^3+1), with x > -1
a) Describe the behavior of g(x) as x--> infinity
b) Describe the beavior of g(x) as x--> -1^+
c) For what values of x is the function increasing? Decreasing? Neither?
d) Find, if any, the local max and local min values of t.
e) For what values of x is the function concave up? COncave down? Neither?
f) Find, if any, the inflection points of f.
Explanation / Answer
1) a) f(x) = e^(1/x) x tends to infinity then (1/x) tends to 0 so e^(1/x)=1 so lim x tends to 0 is f(x) = e^(1/x)=1 f(x) = e^(1/x) x tends to negative infinity then (1/x) tends to 0 so e^(1/x)=1 so lim x tends to 0 is f(x) = e^(1/x)=1 b) f(x) as x--> 0+, 1/ x tends to infinity so f(x)=e^(1/x) is infinty and as x--> 0-; that is, lim f(x) as x--> 0-. 1/ x tends to negative infinity so f(x)=e^(1/x) is negative infinty c) 1/x tends to be infinity at 0 so f value is infinity as described above so at x it has a asymptode d) from 0+ to infinity it is decreasing and from 0- to nagative infinity it is increasing e) from 0+ to infinity it is concave up 0- to nagative infinity it is concave down f) there is no point of inflation at x=0 we have point of undulation A point where the curvature vanishes but does not change 2) a) when x--> infinity and x is>-1 x^3 is always > -a so, from x is (-1,0] it is (-infinity,0] and from x (0, infinity) its range is also (0, infinity) b) question is not clear if it is at -1+ then function comes -infinity c) for the domain (-1,infinity) range is (-infinity , +infinity) and for all the value of domain it is continuous and increasing d) since the nature of the curve is not changing there id no local maximum or local minimum e) curve nature is not changing in the entire domain, and concave is downwards for all the values in domain f) since there is no change in curve no inflection points exist.
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