This function looks a bit more complicated than in the previous example, but the work required is very similar. We'll start by factoring the numerator and denominator. {{MATH_123}} The numerator {{MATH_124}} and the denominator {{MATH_125}} are each continuous on {{MATH_126}}. That means the only possibilities to check for vertical asymptotes are those numbers making the denominator zero. Using limits, we must investigate what happens with {{MATH_127}} when {{MATH_128}} and {{MATH_129}}, since {{MATH_130}} and {{MATH_131}} are the only zeroes of the denominator (since {{MATH_132}} is never zero). We write {{ENV_133}} Since {{MATH_134}}, there is no vertical asymptote at {{MATH_135}}. We turn our attention to the limit as {{MATH_136}} approaches {{MATH_137}}. Let us start with the limit from the right. {{MATH_138}} This is a one-sided limit with form {{CMD_139}}. We will determine the sign of the fraction for {{MATH_140}} sufficiently close to, but greater than, {{MATH_141}}. For the numerator, notice that {{MATH_142}}. That means {{MATH_143}} is the sine value of something in Quadrant III, making it negative. That {{MATH_144}} is near {{MATH_145}} makes the {{MATH_146}} factor in the numerator positive. The numerator is negative for all {{MATH_147}} sufficiently close to, but greater than, {{MATH_148}}. For the denominator, {{MATH_149}} is always positive. Since {{MATH_150}}, the factor {{MATH_151}} is also positive. The denominator is positive for all relevant values of {{MATH_152}}. That makes the fraction {{MATH_153}} negative valued for all {{MATH_154}} sufficiently close to, but greater than, {{MATH_155}}. This means {{MATH_156}}. There is a vertical asymptote at {{MATH_157}}. There are no others, since the only other discontinuity is at {{MATH_158}} but {{MATH_159}}. {{CMD_160}}