Find the limit if it exists. If it does hot cxíst explain why Ih the case of ihf

Question

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 1

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

Answer 2

Jeffrey Jordon

Answered 2022-06-29 Author has 2313 answers

Answer is given below (on video)

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Find the limit if it exists. If it does hot cxíst explain why Ih the case of ihf

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