Question of the day

Calculus Level 4

Let $f:\mathbb{R} \to \mathbb{R}$ is defined by $f(x)=\left( x-6 \right)^x$ .

Find the value of $\large \lim_{x \rightarrow 4} f(x).$

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Limit does not exist -16 16 It diverges to infinity

1 solution

Prakash Chandra Rai
Dec 12, 2014

Let y=(given function), taking log both sides, we get ln y = x*ln|(x-6)|, since LHS must be positive, we get domain of f(x) as R - (-6,6). Since, f(x) doesn't exist in neighborhood of 4, hence limit doesn't exists.

You need to specify the co-domain over which you ask the limit. It does exist over $\mathbb{C}$ , and it's value is $16$ .

Pratik Shastri - 6 years, 6 months ago

No need to specify domain

Kundan Patil - 6 years, 6 months ago

I said co-domain, not domain. If complex numbers are permitted, the limit does exist.

Pratik Shastri - 6 years, 6 months ago

@Pratik Shastri – I didn't know about complex analysis. Can you forward me link to some notes, or if you know can you write a note about it. I answered it considering Real system only.

Prakash Chandra Rai - 6 years, 5 months ago

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