Resolved: "Find the following limits or state that they..."

melodykap

Answered question

2021-10-20

Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers.

lim_{x \to 4} \frac{31 x - 42 \sqrt{x + 5}}{3 - \sqrt{x + 5}}

jlo2niT

Skilled2021-10-21Added 96 answers

Find the limit

lim_{x \to 4} [\frac{31 x - 42 \sqrt{x + 5}}{3 - \sqrt{x + 5}}]

First find L.H.L

lim_{x \to 4^{-}} [\frac{31 x - 42 \sqrt{x + 5}}{3 - \sqrt{x + 5}}]

= lim_{h \to 0} [\frac{31 (4 - h) - 42 \sqrt{(4 - h) + 5}}{3 - \sqrt{(4 - h) + 5}}]

= lim_{h \to 0} [\frac{124 - 31 h - 42 \sqrt{4 - h + 5}}{3 - \sqrt{4 - h + 5}}]

= lim_{h \to 0} [\frac{124 - 31 h - 42 \sqrt{9 - h}}{3 - \sqrt{9 - h}}]

= - \infty

lim_{x \to 4^{-}} [\frac{31 x - 42 \sqrt{x + 5}}{3 - \sqrt{x + 5}}] = - \infty

Now find R.H.L

lim_{x \to 4^{+}} [\frac{31 x - 42 \sqrt{x + 5}}{3 - \sqrt{x + 5}}]

= lim_{h \to 0} [\frac{31 (4 + h) - 42 \sqrt{(4 + h) + 5}}{3 - \sqrt{(4 + h) + 5}}]

= lim_{h \to 0} [\frac{124 + 31 h - 42 \sqrt{4 + h + 5}}{3 - \sqrt{4 + h + 5}}]

= lim_{h \to 0} [\frac{124 + 31 h - 42 \sqrt{9 + h}}{3 - \sqrt{9 + h}}]

= \infty

lim_{x \to 4^{+}} [\frac{31 x - 42 \sqrt{x + 5}}{3 - \sqrt{x + 5}}] = \infty

L . H . S \neq R . H . S

This is helpful 0 Flag

Jeffrey Jordon

Expert2022-07-03Added 2605 answers

Answer is given below (on video)

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