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Evaluation Techniques of Limits

L'Hospital's Rule​

L'Hospital's Rule applies to indeterminate forms such as 00\frac{0}{0} or ∞∞\frac{\infty}{\infty}. If we have:

lim⁡x→af(x)g(x)=00orlim⁡x→af(x)g(x)=∞∞\lim_{{x \to a}} \frac{f(x)}{g(x)} = \frac{0}{0} \quad \text{or} \quad \lim_{{x \to a}} \frac{f(x)}{g(x)} = \frac{\infty}{\infty}

then we can evaluate this limit by taking the derivative of the numerator and the derivative of the denominator:

lim⁡x→af(x)g(x)=lim⁡x→af′(x)g′(x)\lim_{{x \to a}} \frac{f(x)}{g(x)} = \lim_{{x \to a}} \frac{f'(x)}{g'(x)}

provided that the result is not another indeterminate form.

Polynomials at Infinity​

To evaluate the limit at infinity of a ratio of polynomials p(x)p(x) and q(x)q(x):

  1. Factor the largest power of xx out of q(x)q(x).
  2. Simplify the expression.
  3. Compute the limit of the simplified expression as xx approaches infinity.

Example:

lim⁡x→∞3x2−45x−2x2=lim⁡x→∞x2(3−4x25x−2)=lim⁡x→∞3−4x25x−2=32\lim_{{x \to \infty}} \frac{3x^2 - 4}{5x - 2x^2} = \lim_{{x \to \infty}} x^2 \left( \frac{3 - \frac{4}{x^2}}{\frac{5}{x} - 2} \right) = \lim_{{x \to \infty}} \frac{3 - \frac{4}{x^2}}{\frac{5}{x} - 2} = \frac{3}{2}

Rationalize Numerator/Denominator​

For limits involving square roots, rationalize either the numerator or the denominator to facilitate the limit evaluation.

Example:

lim⁡x→93−xx−9=lim⁡x→9(3−x)(3+x)(x−9)(3+x)=lim⁡x→99−x(x−9)(3+x)=−1108\lim_{{x \to 9}} \frac{3 - \sqrt{x}}{x - 9} = \lim_{{x \to 9}} \frac{(3 - \sqrt{x})(3 + \sqrt{x})}{(x - 9)(3 + \sqrt{x})} = \lim_{{x \to 9}} \frac{9 - x}{(x - 9)(3 + \sqrt{x})} = -\frac{1}{108}

Piecewise Function​

For piecewise functions, compute the limit from both sides if necessary.

Example:

Let

g(x)={x2+5x<−21−3xx≥−2g(x) = \left\{ \begin{array}{ll} x^2 + 5 & \quad x < -2 \\ 1 - 3x & \quad x \geq -2 \end{array} \right.

Compute two one-sided limits:

lim⁡x→−2−g(x)=x2+5=9\lim_{{x \to -2^-}} g(x) = x^2 + 5 = 9 lim⁡x→−2+g(x)=1−3x=7\lim_{{x \to -2^+}} g(x) = 1 - 3x = 7

Combine Rational Expressions​

To evaluate limits involving rational expressions where variables approach zero:

  1. Combine the rational expressions into a single fraction.
  2. Factor and simplify if possible.
  3. Substitute the approaching value and evaluate the limit.

Example:

lim⁡h→0(1x+h−1x)=lim⁡h→0x−(x+h)x(x+h)=lim⁡h→0−hx(x+h).\lim_{{h \to 0}} \left( \frac{1}{x + h} - \frac{1}{x} \right) = \lim_{{h \to 0}} \frac{x - (x + h)}{x(x + h)} = \lim_{{h \to 0}} \frac{-h}{x(x + h)}.

Simplify and evaluate the limit:

lim⁡h→0−1x(x+h)=−1x2.\lim_{{h \to 0}} \frac{-1}{x(x + h)} = \frac{-1}{x^2}.

These techniques streamline the evaluation process for different types of limit problems.

References​