Q. 4 Explain graphically why it makes... [FREE SOLUTION]

Chapter 3: Q. 4 (page 310)

Explain graphically why it makes sense that a limit $\lim_{x 鈫� 鈭�} \frac{f (x)}{g (x)}$ of the indeterminate form $\frac{鈭�}{鈭�}$ would be related to $\lim_{x 鈫� 鈭�} \frac{f' (x)}{g' (x)}$ .

Short Answer

Expert verified

Limits of the indeterminate form $\frac{鈭�}{鈭�}$ are related to the limit of the derivatives of its numerator and denominator functions.

Step by step solution

Step 1. Given information

We have to prove graphically that a limit $\lim_{x 鈫� 鈭�} \frac{f (x)}{g (x)}$ of the indeterminate form $\frac{鈭�}{鈭�}$ would be related to $\lim_{x 鈫� 鈭�} \frac{f' (x)}{g' (x)}$ .

Step 2. Proof of the given question graphically.

Let us take an example

$\lim_{x 鈫� 鈭�} \frac{1 - 2 x^{2}}{2 - 3 x^{2}}$

The value of the limit is in the form of $\frac{鈭�}{鈭�}$ .

The graph of $f (x) = 1 - 2 x^{2}$ and $g (x) = 2 - 3 x^{2}$ are

As $x 鈫� 鈭�$ , the graph of both the functions become vertical same as the vertical tangent at $x = 鈭�$ . So, $\lim_{x 鈫� 鈭�} \frac{f (x)}{g (x)}$ of the indeterminate form $\frac{鈭�}{鈭�}$ would be related to $\lim_{x 鈫� 鈭�} \frac{f (x)}{g (x)}$ .

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91影视

Explain graphically why it makes sense that a limit $\lim_{x 鈫� 鈭�} \frac{f (x)}{g (x)}$ of the indeterminate form $\frac{鈭�}{鈭�}$ would be related to $\lim_{x 鈫� 鈭�} \frac{f' (x)}{g' (x)}$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Proof of the given question graphically.

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Most popular questions from this chapter

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91影视

Explain graphically why it makes sense that a limit limx鈫�鈭�fxgxof the indeterminate form 鈭�鈭� would be related to limx鈫�鈭�f'xg'x.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Proof of the given question graphically.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Statistics

Decision Maths

Pure Maths

Probability and Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Explain graphically why it makes sense that a limit $\lim_{x 鈫� 鈭�} \frac{f (x)}{g (x)}$ of the indeterminate form $\frac{鈭�}{鈭�}$ would be related to $\lim_{x 鈫� 鈭�} \frac{f' (x)}{g' (x)}$ .