Calculus: Finding x, Derivative and Limit

Question

1) Find x if e^5-6x = 5.

2) Find derivative of g(x) = e^x √x with respect to x.

3) Evaluate following limit:

lim ln(sin(x))

x -> 0+

Answer 1

1) This type of problem can be solved via natural logarithms (natural because that is what e is related to)

e^(5-6x) = 5

ln(e^(5-6x)) = ln(5)

5-6x = ln(5)

x = -(ln(5) - 5)/6

= (5 - ln(5))/6

= (5 - 1.609)/6

= 0.565

.

2) If f(x) = u(x)*v(x), then f'(x) = u(x)*v'(x) + u'(x)*v(x)

In this case, u(x) = e^x and v(x) = sqrt(x) = x^(1/2)

u'(x) = e^x and v'(x) = 1/2 * x^(-1/2) = 1/(2*sqrt(x))

f'(x) = e^x/2*sqrt(x) + e^x * sqrt(x)

= e^x(1/2sqrt(x) + sqrt(x)) which can be factored to be

= e^x*sqrt(x) * (1/x + x)

3) lim as x->0+ of ln(sin(x)); note that sin(0) = 0, and ln(0) = - infiniti (in other words, e^(-infinity)=0)

This is a composite function, f(g(x)). In such a case if lim g(x) = b as x->a, then the limit of f is f(b) as x->a. Therefore

lim (sin(x) = 0

x->0+

lim ln(sin(x)) = lim ln(0) = - infinity

x->0 x->0

0+ was chosen in the problem rather than 0 so that the limit could have a finite value, but as 0+ gets infinitesimally small, ln(0+) gets increasingly large without bound.