Question

Certainly, here are the questions in plain text format:

Find lim (x → 4) (x - 4) / (sqrt(x) - 2).

Find lim (x → -5+) x(1 + sqrt(5 + x)).

Answer 1

The student has inquired about calculating two limits: one that involves a zero/zero indeterminate form and another straightforward substitution as x approaches a value from the right. By algebraic manipulation for the first and direct substitution for the second, the limits are found to be 4 and 0, respectively.

Step-by-step explanation:

The student has asked two questions related to limits in mathematics:

Question 1:

Find lim (x → 4) (x - 4) / (√x - 2). In this question, we encounter a zero/zero indeterminate form when we plug in x = 4 directly into the expression. To solve this, we need to manipulate the expression to eliminate the indeterminate form. Notice that x - 4 can be written as (√x)2 - 22. We can then factor it as (√x + 2)(√x - 2) and cancel the (√x - 2) term with the denominator. This simplification leaves us with √x + 2, and we can then evaluate the limit by plugging in x = 4, yielding an answer of 4.

Question 2:

Find lim (x → -5+) x(1 + √(5 + x)). For this limit, we're approaching -5 from the right. There's no indeterminate form, so we can simply substitute x = -5 into the expression, which gives us the value 0, since x(1 + √(5 + x)) becomes 0(1 + √(0)), which is 0.