Question

Given that \(\mathop {\lim }\limits_{x \to a} f\left( x \right) = 0\)

, \(\mathop {\lim }\limits_{x \to a} g\left( x \right) = 0\), \(\mathop {\lim }\limits_{x \to a} h\left( x \right) = 1\), \(\mathop {\lim }\limits_{x \to a} p\left( x \right) = \infty \) and \(\mathop {\lim }\limits_{x \to a} q\left( x \right) = \infty \)

which of the following limits are indeterminate forms? For any limit that is not an indeterminate form, evaluate it where possible.

3.

1. \(\mathop {\lim }\limits_{x \to a} \left( {f\left( x \right) - p\left( x \right)} \right)\)
2. \(\mathop {\lim }\limits_{x \to a} \left( {p\left( x \right) - q\left( x \right)} \right)\)
3. \(\mathop {\lim }\limits_{x \to a} \left( {p\left( x \right) + q\left( x \right)} \right)\)

Short answer

1. The value of limit is \( - \infty \).
2. Indeterminate form of type \(\infty - \infty \).
3. The value of limit is \(\infty \).

Step-by-step solution

 Indeterminate form

If the value of the limit exists in the form of type \(\frac{0}{0}\), \(\frac{\infty }{\infty }\) or \(\infty - \infty \), then it is referred to as indeterminate form.

(a) Step 2: Evaluation of limits

Calculate the limits by using given values.

\(\begin{array}{c}\mathop {\lim }\limits_{x \to a} \left( {f\left( x \right) - p\left( x \right)} \right) = \mathop {\lim }\limits_{x \to a} f\left( x \right) - \mathop {\lim }\limits_{x \to a} p\left( x \right)\\\mathop {\lim }\limits_{x \to a} \left( {f\left( x \right) - p\left( x \right)} \right) = 0 - \infty \\\mathop {\lim }\limits_{x \to a} \left( {f\left( x \right) - p\left( x \right)} \right) = - \infty \end{array}\)

Hence, \(\mathop {\lim }\limits_{x \to a} \left( {f\left( x \right) - p\left( x \right)} \right) = - \infty \).

(b) Step 3: Evaluation of limits

Calculate the limits by using given values.

\(\begin{array}{c}\mathop {\lim }\limits_{x \to a} \left( {p\left( x \right) - q\left( x \right)} \right) = \mathop {\lim }\limits_{x \to a} p\left( x \right) - \mathop {\lim }\limits_{x \to a} q\left( x \right)\\\mathop {\lim }\limits_{x \to a} \left( {p\left( x \right) - q\left( x \right)} \right) = \infty - \infty \end{array}\)

Hence, \(\mathop {\lim }\limits_{x \to a} \left( {p\left( x \right) - q\left( x \right)} \right)\) is an indeterminate form of the type \(\infty - \infty \).

(c) Step 4: Evaluation of limits

Calculate the limits by using given values.

\(\begin{array}{c}\mathop {\lim }\limits_{x \to a} \left( {p\left( x \right) + q\left( x \right)} \right) = \mathop {\lim }\limits_{x \to a} p\left( x \right) + \mathop {\lim }\limits_{x \to a} q\left( x \right)\\\mathop {\lim }\limits_{x \to a} \left( {p\left( x \right) + p\left( x \right)} \right) = 0 + \infty \\\mathop {\lim }\limits_{x \to a} \left( {p\left( x \right) + q\left( x \right)} \right) = \infty \end{array}\)

Hence, \(\mathop {\lim }\limits_{x \to a} \left( {p\left( x \right) + q\left( x \right)} \right) = \infty \).