Key to "Find the limit if it exists. If it..."

arenceabigns

Answered question

2021-10-19

Find the limit if it exists. If it does hot cxíst explain why Ih the case of ihfinite limits, malk suve that you check both the left and right limit at the point in the question.

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

Answer & Explanation

gotovub

Skilled2021-10-20Added 98 answers

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

L . H . L = lim_{h \to 0} \frac{({(1 - h)}^{2} - 1) e^{\frac{1}{x - h - x}}}{(x - h - x)}

= lim_{h \to 0} \frac{x + h^{2} - 2 h - x}{- h} e^{\frac{1}{h}}

= lim_{h \to 0} - (h - 2) e^{- \frac{1}{h}}

Taking limit:

L . H . L = 0

R . H . L . = lim_{h \to 0} \frac{({(1 + h)}^{2} - 1) e^{\frac{1}{x + h - x}}}{x + h - x}

= lim_{h \to 0} \frac{x + h^{2} + 2 h - x}{h} e^{\frac{1}{h}}

= lim_{h \to 0} (h + 2) e^{\frac{1}{h}}

Taking limit

R . H . L = \infty

(does not exist finity)

\Rightarrow L . H . L \neq R . H . L

\Rightarrow lim_{x \to 1} \frac{(x^{2} - 1) e^{\frac{1}{x - 1}}}{(x - 1)}

does not exist

Jeffrey Jordon

Expert2022-06-29Added 2605 answers

Answer is given below (on video)

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