How do you evaluate #[ ( x )^(2x) ]# as x approaches 0+?

Answer 1

I don't think this can be done algebraically. I think you'll need l"Hopital's rule.

#x^(2x) = e^(2xlnx)#
Since the exponential function is continuous, we can simplify the problem to find #lim_(xrarr0^+)xlnx = L#. (The limit we want will then be #e^(2L)#.)
#lim_(xrarr0^+)xlnx# has initial form #0*(-oo)#. This form is indeterminate.
We can apply l'Hopital''s Rule if we can make the form #0/0# or #+-oo/oo#.
#lim_(xrarr0^+)xlnx = lim_(xrarr0^+) lnx/(1/x)# has form #-oo/oo#

Applying the rule gets us

# = lim_(xrarr0^+) (1/x)/(-1/x^2) = lim_(xrarr0^+)( -x) = 0 #.
Therefore, #lim_(xrarr0^+)(2xlnx) = 0#

We conclude that

#lim_(xrarr0^+) x^(2x) = lim_(xrarr0^+) e^(2xlnx)#
# = e^((lim_(xrarr0^+) 2xlnx))#
# = e^0 = 1#
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Answer 2

To evaluate [ ( x )^(2x) ] as x approaches 0+, we can use the limit definition. Taking the natural logarithm of the expression, we have ln[( x )^(2x)]. Applying the properties of logarithms, this simplifies to 2x * ln(x). Now, we can evaluate the limit of this expression as x approaches 0+. Using L'Hôpital's rule, we differentiate the numerator and denominator with respect to x. The derivative of 2x is 2, and the derivative of ln(x) is 1/x. Taking the limit as x approaches 0+, we get 2 * (1/0), which is undefined. Therefore, the limit of [ ( x )^(2x) ] as x approaches 0+ does not exist.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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