Find the limits: lim_{xrightarrow3}frac{-2}{x-3}

Question

Find the limits:

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

Answer 1

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

Expert Answer