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Numerade Educator

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Problem 33 Hard Difficulty

Determine the infinite limit.

$ \displaystyle \lim_{x \to 1}\frac{2-x}{(x-1)^2} $

Answer

As $x \rightarrow 1$, the numerator $(2-x)$ approaches a positive number $1$. At the same time the denominator $(x-1)^2$ is always positive and $(x-1)^2 \rightarrow 0$.

Thus the fraction increases without bounds:

$\lim _{x \rightarrow 1} \frac{2-x}{(x-1)^{2}}=+\infty$

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Video Transcript

Okay in this problem we want to investigate the limit of this function as X approaches one. Uh So we graft we're going to investigate this limit graphically. So, using dez mose uh we graft to function two minus X over x minus one squared. And as X approaches one for the limit to exist, it's going to actually be an infinite limit. So will it be positive infinity or negative infinity for the limit to exist as X approaches one? The limit has to be the same as it approaches one from the negative side. And as it approaches one from the positive side, uh you can see that as X approaches one from the negative side from the left, the limit of the function is going to be positive infinity because as you get closer and closer to X equals one. Uh the function is increasing without bound is getting higher and higher and higher. Uh Likewise, the same story as X approaches one from the positive side. Uh the function is increasingly getting higher and higher, approaching positive infinity. Um The closer you get to one from the positive side, So the limit as x approaches one of this function f of X is positive infinity.