Determine the one-sided limits numerically or graphically. If infinite, state wh

Question

Determine the one-sided limits numerically or graphically. If infinite, state whether the one-sided limits are ∞ or −∞, and describe the corresponding vertical asymptote.

lim_{x \to - 2^{\pm}} \frac{4 x^{2} + 7}{x^{3} + 8}

Answer 1

Given function is:

f (x) = \frac{4 x^{2} + 7}{x^{3} + 8}

lim_{x \to - 2^{+}} (\frac{4 x^{2} + 7}{x^{3} + 8}) = \frac{lim_{x \to - 2^{+}} (4 x^{2} + 7)}{lim_{x \to - 2^{+}} (x^{3} + 8)}

Put, x=-2

= \frac{4 {(- 2)}^{2} + 7}{{(- 2)}^{3} + 8}

= \frac{4 \times 4 + 7}{- 8 + 8}

= \frac{23}{0}

= \infty

and

lim_{x \to - 2^{-}} (\frac{4 x^{2} + 7}{x^{3} + 8}) = \frac{lim_{x \to - 2^{-}} (4 x^{2} + 7)}{lim_{x \to - 2^{-}} (x^{3} + 8)}

Put, x=-2

= \frac{4 {(- 2)}^{2} + 7}{{(- 2)}^{3} + 8}

= \frac{4 \times 4 + 7}{- 8 + 8}

= \frac{23}{0}

= - \infty

∴ x = - 2

is vertical asymptote.

Determine the one-sided limits numerically or graphically. If infinite, state wh

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