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One-sided infinite limit at x = 3

Consider the function f(x)=49−x2f(x) = \dfrac{4}{9 - x^2}, which has vertical asymptotes where its denominator equals zero.

Evaluate the one-sided limit lim⁡x→3−49−x2\lim_{x \to 3^{-}} \frac{4}{9 - x^2} State whether the limit is +∞+\infty, −∞-\infty, or a finite value.

Hint

Factor 9−x2=(3−x)(3+x)9 - x^2 = (3-x)(3+x) and examine the sign of each factor as xx approaches 3 from the left. Since the numerator is a positive constant, the sign of the denominator determines the direction of divergence.

Step-by-step solution

Factor the denominator as 9−x2=(3−x)(3+x)9 - x^2 = (3-x)(3+x). As x→3−x \to 3^{-}, we have xx slightly less than 3, so 3−x3 - x is a small positive number, while 3+x3 + x approaches 6, a positive number. Thus the denominator (3−x)(3+x)(3-x)(3+x) approaches 00 through positive values.

Since the numerator is the constant 4>04 > 0 and the denominator approaches 0+0^{+}, the fraction 49−x2\frac{4}{9-x^2} grows without bound in the positive direction. Therefore lim⁡x→3−49−x2=+∞.\lim_{x \to 3^{-}} \frac{4}{9 - x^2} = +\infty.

One-sided infinite limit at x = -3

Evaluate the one-sided limit lim⁡x→−3+49−x2\lim_{x \to -3^{+}} \frac{4}{9 - x^2} State whether the limit is +∞+\infty, −∞-\infty, or a finite value.

Hint

Factor 9−x2=(3−x)(3+x)9 - x^2 = (3-x)(3+x) and check the sign of each factor as xx approaches −3-3 from the right. The factor (3+x)(3+x) is the one approaching zero; determine its sign to find the direction of divergence.

Step-by-step solution

Factor the denominator as 9−x2=(3−x)(3+x)9 - x^2 = (3-x)(3+x). As x→−3+x \to -3^{+}, xx is slightly greater than −3-3. The factor (3+x)(3+x) is a small positive number (approaching 0+0^+), while 3−x3 - x approaches 3−(−3)=63 - (-3) = 6, a positive number. Thus the denominator (3−x)(3+x)(3-x)(3+x) approaches 00 through positive values.

Since the numerator is the constant 4>04>0 and the denominator approaches 0+0^{+}, the fraction becomes arbitrarily large and positive. Therefore lim⁡x→−3+49−x2=+∞.\lim_{x \to -3^{+}} \frac{4}{9 - x^2} = +\infty.

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