Question

Determine the following limits.b. lim x→4^− x − 5 / (x − $$4)^2$$

Accepted Answer

Step 1: Identify the type of limit problem. This is a one-sided limit as $$ x $$ approaches 4 from the left (denoted by $$ x 	o 4^- ). $$Step 2: Substitute $$ x = 4 $$ into the expression $$ \frac{x - 5}{(x - 4)^2}  to $$check for any indeterminate forms. You will find that the numerator becomes $$ 4 - 5 = -1 $$ and the denominator becomes $$ (4 - 4)^2 = 0 $$, indicating a division by zero, which suggests a potential vertical asymptote. Step 3: Analyze the behavior of the function as $$ x $$ approaches 4 from the left. Since $$ x 	o 4^- $$, $$ x  is $$slightly less than 4, making $$ x - 4  a $$small negative number. Therefore, $$ (x - 4)^2  is a $$small positive number. Step 4: Consider the sign of the expression. The numerator $$ x - 5  is $$negative when $$ x  is $$slightly less than 4, and the denominator $$ (x - 4)^2  is $$positive. Thus, the overall expression $$ \frac{x - 5}{(x - 4)^2}  is $$negative. Step 5: Conclude the behavior of the limit. As $$ x 	o 4^- $$, the expression $$ \frac{x - 5}{(x - 4)^2} $$ approaches negative infinity because the numerator is negative and the denominator approaches zero from the positive side.

Determine the following limits.

b. lim x→4^− x − 5 / (x − 4)^2

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Limits

One-Sided Limits

Indeterminate Forms

Determine the following limits.b. lim x→4^− x − 5 / (x − 4)^2