Math Problem Statement

lim (h → 0) [(√(49 + h) - 7) / h]

Solution

The given limit is:

limh049+h7h\lim_{{h \to 0}} \frac{\sqrt{49 + h} - 7}{h}

To evaluate this limit, we can use the technique of multiplying by the conjugate to simplify the expression. Let's rewrite the expression:

limh049+h7h49+h+749+h+7\lim_{{h \to 0}} \frac{\sqrt{49 + h} - 7}{h} \cdot \frac{\sqrt{49 + h} + 7}{\sqrt{49 + h} + 7}

Now, the numerator becomes a difference of squares:

=limh0(49+h)272h(49+h+7)= \lim_{{h \to 0}} \frac{(\sqrt{49 + h})^2 - 7^2}{h (\sqrt{49 + h} + 7)}

Simplify the squares:

=limh049+h49h(49+h+7)= \lim_{{h \to 0}} \frac{49 + h - 49}{h (\sqrt{49 + h} + 7)}

This reduces to:

=limh0hh(49+h+7)= \lim_{{h \to 0}} \frac{h}{h (\sqrt{49 + h} + 7)}

Cancel the hh terms:

=limh0149+h+7= \lim_{{h \to 0}} \frac{1}{\sqrt{49 + h} + 7}

Now, as h0h \to 0, we substitute h=0h = 0:

=149+0+7=17+7=114= \frac{1}{\sqrt{49 + 0} + 7} = \frac{1}{7 + 7} = \frac{1}{14}

Thus, the value of the limit is:

114\boxed{\frac{1}{14}}

Would you like further details on this process or have any questions?

Here are some related questions for further exploration:

  1. What is the technique of using conjugates, and how is it applied in other limits?
  2. How do you evaluate limits that involve radicals without using the conjugate method?
  3. What is L'Hopital's rule, and could it be applied to solve this limit?
  4. How does continuity of functions impact the evaluation of limits?
  5. How are limits with trigonometric functions solved using similar techniques?

Tip: When facing limits involving square roots, multiplying by the conjugate can often help simplify the expression.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Limits
Radicals
Algebraic Simplification

Formulas

Difference of squares
Limit of a function as h approaches 0

Theorems

Limit definition
Conjugate multiplication method

Suitable Grade Level

Grades 11-12