Best solutions to - "Find the limits: limx→3−2x−3"

Tahmid Knox

Answered question

2020-12-27

Find the limits:

lim_{x \to 3} \frac{- 2}{x - 3}

Obiajulu

Skilled2020-12-28Added 98 answers

Given:

lim_{x \to 3} (- \frac{2}{x - 3})

LEFT-HAND LIMIT
Apply the constant multiple rule

lim_{x \to 3^{-}} c f (x) = c \cdot lim_{x \to 3^{-}} f (x) w i t h c = - 2 a n d f (x) = \frac{1}{x - 3} :

lim_{x \to 3^{-}} (- \frac{2}{x - 3}) = (- 2 lim_{x \to 3^{-}} \frac{1}{x - 3})

The function decreases without a bound:

lim_{x \to 3^{-}} \frac{1}{x - 3} = - \infty

Therefore,

lim_{x \to 3^{-}} (- \frac{2}{x - 3}) = \infty

RIGHT-HAND LIMIT
Apply the constant multiple rule

lim_{x \to 3^{+}} c f (x) = c \cdot lim_{x \to 3^{+}} f (x) w i t h c = - 2 a n d f (x) = \frac{1}{x - 3} :

lim_{x \to 3^{+}} (- \frac{2}{x - 3}) = (- 2 lim_{x \to 3^{+}} \frac{1}{x - 3})

The function grows without a bound:

lim_{x \to 3^{+}} \frac{1}{x - 3} = \infty

Therefore,

lim_{x \to 3^{+}} (- \frac{2}{x - 3}) = - \infty

Answer:

lim_{x \to 3} (- \frac{2}{x - 3})

does not exist, since the corresponding one-sided limits are not equal:

lim_{x \to 3^{-}} (- \frac{2}{x - 3}) = \infty, lim_{x \to 3^{+}} (- \frac{2}{x - 3}) = - \infty

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