Please solve all Evaluate lim x rightarrow infinity f (x) and lim x rightarrow -

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Evaluate lim x rightarrow infinity f (x) and lim x rightarrow -infinity f (x) for the following function. Use infinity or - infinity where appropriate. Then give the horizontal asymptote(s) of f (if any). For the function find the following. Evaluate lim x rightarrow infinity f (x) and lim x rightarrow -infinity f (x) and then identify the horizontal asymptotes. Find the vertical asymptote. For the vertical asymptote x = a, evaluate lim x rightarrow f(x) and lim x rightarrow f (x).

Explanation / Answer

Q1.

(x)=(x^3+4)/(2x^3+sqrt(16x^6+3))=(1+4/x^3)/(2+sqrt(16x^6+3)/x^3)

case 1:

x tends to infinity:

x>0

Hence,

f(x)=(x^3+4)/(2x^3+sqrt(16x^6+3))=(1+4/x^3)/(2+sqrt(16+3/x^6))=(1+0)/(2+4)=1/6(ans)

x tends to -infinity:

x<0

Hence,

f(x)=(x^3+4)/(2x^3+sqrt(16x^6+3))=(1+4/x^3)/(2-sqrt(16+3/x^6))=(1+0)/(2-4)=-1/2(ans)

since root(a)/b=-root(a/b^2) when b<0

Q2.

f(x)=(sqrt(x^2+3x+5)-3)/(x-1)=(sqrt(1+3/x+5/x^2)-3/x)/(1-1/x)) when x>0

=(-sqrt(1+3/x+5/x^2)-3/x)/(1-1/x)) when x<0

when x tends to infinity:

sqrt(1)/1=1(ans)

when x tends to -infinity:

-sqrt(1)/1=-1(ans)

hence horizontal assymtotes are y=1 and y=-1

vertical assymtote can be x=1 (since dinominator not equal to 0)

for that we use the l hopitals rule:

we find LHL=RHL=5/6

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