Approximating Limits Graphically

In this video, Mr. Pacquiao demonstrates how to approximate limits graphically using a calculator. He explains the process with two examples: the limit of (1+x)^(1/x) as x approaches 0, and the limit of sin(x)/x as x approaches 0. For each example, he shows how to use a graph to estimate the limit when direct substitution is not possible. The video emphasizes the importance of understanding graphical representations to solve limits that are difficult to evaluate algebraically.

What is the approximate value of the limit of (1+x)^(1/x) as x approaches 0?

• Approximately 1

• Approximately 0

• Approximately 3.14

• Approximately 2.718

What does the presence of a hole in the graph indicate?

• The function is not continuous at that point.

• The limit does not exist.

• The limit is zero.

• The function is perfectly continuous.

What is the significance of the value 2.718 in the context of limits?

• It is an arbitrary constant.

• It is the golden ratio.

• It is the value of pi.

• It is the base of the natural logarithm.

Why can't we directly substitute 0 in the expression (1+x)^(1/x)?

• It simplifies the expression too much.

• It results in an undefined expression.

• It cancels out the x variable.

• It makes the limit equal to 1.

Why do we use a graphical approach to approximate limits?

• It's more accurate than algebraic methods.

• It's the only way to find limits.

• Some limits are difficult to evaluate algebraically.

• It requires less calculation.

Why is direct substitution not always effective in finding limits?

• It always results in a value of 0.

• It changes the original function.

• It simplifies the function too much.

• It can lead to indeterminate forms.

What is the approximate value of the limit of sin(x)/x as x approaches 0?

• Approximately 0

• Approximately 1

• Approximately 2.718

• Approximately 0.999

Why must the calculator be in radian mode to graph sin(x)/x?

• Because the limit only exists in radians.

• To ensure the graph is scaled correctly.

• To accurately represent the trigonometric function.

• Because sin(x) is undefined in degree mode.

What does approaching a limit from both sides help determine?

• The maximum value of the function.

• Whether the limit exists and its value.

• The slope of the tangent at that point.

• The minimum value of the function.

What does a y-value getting closer to 1 as x approaches 0 indicate about the limit?

• The limit does not exist.

• The limit is approximately 1.

• The limit approaches infinity.

• The limit is undefined.