Question

Find all vertical asymptotes x=ax=a of the following functions. For each value of aa, determine lim⁡x→a+f(x){\displaystyle\lim_{x	o $$a^{+}$$}}f\left(xight), lim⁡x→a−f(x){\displaystyle\lim_{x	o $$a^{-}$$}}f\left(xight), and lim⁡x→af(x){\displaystyle\lim_{x	o a}}f\left(xight).f(x)=cos⁡(x)x2+2xf\left(xight)=\frac{\cos\left(xight)}{$$x^2+2x$$}

Accepted Answer

Step 1: Identify the points where the denominator of the function is zero, as these are potential vertical asymptotes. For the function $$ f(x) = \frac{\cos(x)}{x^2 + 2x} $$, set the denominator equal to zero: $$ x^2 + 2x = 0 . $$Step 2: Solve the equation $$ x^2 + 2x = 0  by $$factoring. Factor out an $$ x  to $$get $$ x(x + 2) = 0 . $$This gives the solutions $$ x = 0 $$ and $$ x = -2 . $$Step 3: Determine the behavior of the function as $$ x $$ approaches each of these values from the left and right. For $$ x = 0 $$, evaluate $$ \lim_{x 	o 0^+} f(x) $$ and $$ \lim_{x 	o 0^-} f(x) . $$Step 4: Similarly, evaluate the limits for $$ x = -2 . $$Calculate $$ \lim_{x 	o -2^+} f(x) $$ and $$ \lim_{x 	o -2^-} f(x) . $$Step 5: Analyze the results of these limits. If the one-sided limits approach $$ \pm \infty $$, then $$ x = 0 $$ and $$ x = -2 $$ are vertical asymptotes. If the two-sided limit $$ \lim_{x 	o a} f(x) $$ does not exist or is infinite, it confirms the presence of a vertical asymptote at $$ x = a .$$

Find all vertical asymptotes $x = a$ of the following functions. For each value of $a$ , determine $\lim_{x \to a^{+}} f (x)$ , $\lim_{x \to a^{-}} f (x)$ , and $\lim_{x \to a} f (x)$ .
$f (x) = \frac{\cos (x)}{x^{2} + 2 x}$

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Concetti chiave

Vertical Asymptotes

Limits

Continuous Functions