# math - a The expression x2 1(x 1 explodes at x = 1 because...

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a) The expression, x 2 + 1 / (x – 1) “explodes” at x = 1 because when the denominator of the function is set = 0, it is found that x = 1, so b is also 1. As a result, the limit as x approaches 1 from the left of x 2 + 1 / (x – 1) approaches either infinity or negative infinity. It can thus be found that the limit approaches positive infinity by observing that the numerator, x 2 + 1 would be positive if 1 were substituted for x: 1 2 + 1, so the numerator would be 2 if x were evaluated at 1. Because x approaches b from the left side, the denominator is also positive. Therefore, the division of the numerator, which is positive and denominator, which is positive, makes the expression, x 2 + 1 / (x – 1), positive at x = 1. It can thus be derived that the limit of x 2 + 1 / (x – 1) as x approaches 1 is positive infinity. So b = 1. The expression, cos(x)/(x-2) “explodes” at x = 2 because when the denominator of the function is set = 0, it is found that x = 2, so b is also 2. The limit as x approaches 2 from the left of cos(x) / (x – 2) thus approaches either infinity or negative infinity.

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