How do you find the limit of #(2^w-2)/(w-1)# as w approaches 1?

Answer 1

#ln4#

We want to evaluate this: #lim_(wrarr1)(2^w-2)/(w-1)#
Putting #w=1# directly gives us: #(2^1-2)/(1-1)=0/0# Which is an indeterminate form. So, we a free to apply Hospital's rule
Hospital's rule simply says that we differentiate top and bottom of the #lim# until the indeterminate form disappears.
#lim_(wrarr1)(2^w-2)/(w-1) " "# becomes #" "lim_(wrarr1)(2^wln2)/(1)=2^1ln2=2ln2=ln2^2=color(blue)ln4#
#color(white)#
Suppose, #y=2^w " "# then #" "lny=wln2#
Differentiating implicitly we get, #" "1/y(dy)/(dx)=ln2#
#=>(dy)/(dx)=yln2=2^wln2#
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Answer 2

To find the limit of (2^w-2)/(w-1) as w approaches 1, we can use L'Hôpital's rule. Taking the derivative of the numerator and denominator separately, we get (ln(2)*2^w)/(1). Evaluating this expression at w=1, we have (ln(2)*2^1)/(1) = ln(2)*2. Therefore, the limit of (2^w-2)/(w-1) as w approaches 1 is ln(2)*2.

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Answer 3

To find the limit of ( \frac{2^w - 2}{w - 1} ) as ( w ) approaches ( 1 ), you can use L'Hôpital's Rule or the property of the derivative of ( 2^w ) at ( w = 1 ). Applying L'Hôpital's Rule, differentiate the numerator and denominator separately with respect to ( w ), then evaluate the limit again. Alternatively, you can rewrite ( 2^w ) as ( e^{w \ln(2)} ), then use the limit definition of the derivative of ( e^x ) at ( x = 0 ). Either method yields the result ( \ln(2) ).

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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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